There is a n overriding complication to plotting FRF data which derives from the fact that they are complex and thus there are three quantities - frequency plus two parts of the complex function - and these cannot be fully displayed on a standard x-y graph. Because of this, any such simple plot can only show two of the three quantities and so there are different possibilities available, several of which are used from time to time.
The four most common forms of presentation are:
(i) Modulus (of FRF) vs. Frequency and Phase vs. Frequency (the Bode type of plot, consisting of two graphs);
Real Part (of FRF) vs. Frequency and Imaginary Part vs.
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Fig. 2.6 FRF plots for undamped SDOF system (linear scales).
(a) Receptance FRF; Mobility FRF
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Accelerance FRF
Real Part (of reciprocal of FRF) vs. Frequency (or
and Imaginary Part (of reciprocal of FRF) vs. Frequency; and (iv) Real Part (of FRF) vs. Imaginary Part (of FRF). (The so-called
'Nyquist plot': a single graph which does not contain frequency information explicitly).
We shall now examine the form and use of these types of graphical display and identify the particular advantages or features of each.
(i) A classical Bode plot is shown in Fig. for the receptance of a typical SDOF system without damping. Corresponding plots for the mobility and inertance of the same system are shown in Figs.
and (c), respectively.
One of the problems with these FRF properties, a s with much vibration data, is the relatively wide range of values which must be encompassed no matter which type of FRF is used. In order to cope with this problem, it is often appropriate to make use of logarithmic scales and the three functions specified above have been replotted i n Figs. (b) and (c) using logarithmic scales
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FREQUENCY (Hz)
FREQUENCY (Hz)
Fig. 2.7 FRF plots for system (log-log scales).
(a) Receptance FRF; (b) Mobility FRF
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Fig. 2.7 FRF plots for undamped SDOF system (log-log scales).
(c) Accelerance FRF
for all frequency and modulus axes. The result is something of a transformation in that in each plot can now be divided into three regimes:
a low-frequency straight-line characteristic;
a high-frequency straight-line characteristic, and
the resonant region with its abrupt magnitude and phase variations.
It is helpful and instructive to superimpose on these log-log plots a grid of lines which show the relevant FRF characteristics separately of simple mass elements and simple spring elements.
Table 2.2 shows the corresponding expressions for a,, Y,, etc. and from this it is possible to see that mass and stiffness properties will always appear as straight lines on log (modulus) vs. log (frequency) plots, as shown i n Fig. 2.8. These have i n fact been included i n Fig. 2.7 but their significance can be further appreciated by reference to Fig. 2.8 which shows the mobility modulus plots for two different systems. By referring to and interpolating between the mass- and stiffness-lines drawn on the plot, we can deduce that system (a) behaves a s would a mass of
kg with a spring stiffness of 2.5 while system has
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corresponding values of 0.8 kg and 120 respectively.
Table 2.2 Frequency Responses of Mass and Stiffness Elements
FRF Parameter Mass Stiffness
FREQUENCY (Hz)
Fig. 2.8 Mobility FRF plots for different undamped SDOF systems RECEPTANCE
MOBILITY
ACCELERANCE
This basic style of displaying FRF data applies to all types of system, whether damped or not, while the other forms are only applicable to damped systems and then tend to be sensitive to the type of damping. Fig. 2.9 shows the basic example system plotted for different levels of viscous damping with a zoomed detail of the narrow band around resonance is the only region that the damping has any influence on the FRF plot.
-l/w2m m
log
(ii) Companion plots for Real Part (Re.) vs. Frequency and Imaginary Part vs. Frequency are shown in Fig. 2.10 for the SDOF system with light viscous damping. three forms of the FRF are shown and from these we can see how the phase change through
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- -02/k
2 -
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FREQUENCY ( Hz)
Fig. 2.9 Resonance region detail of FRF plot for damped SDOF system
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Fig. 2.10 Plots of real and imaginary parts of FRF for damped SDOF system.
(a) Receptance; (b) Accelerance
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t h e resonance region is characterised by a sign change i n one part accompanied by a peak or value i n the other part.
It should be noted here t h a t the use of logarithmic scales is not feasible in this case primarily because it is necessary to accommodate both positive and negative values (unlike the modulus plots) and this would be impossible with logarithmic axes. Partly for this reason, and others which become clearer when dealing with MDOF systems, this format of display is not so widely used a s the preceding ones.
The so-called 'inverse' or 'reciprocal' plots are, however, more interesting in that they have the potential of providing rather more insight into the system whose characteristics they represent. First of all, it can be seen the expression for the inverse receptance (see equation (2.15)) that the Real P a r t depends entirely on the mass and stiffness properties while the Imaginary Part is a function only of the damping. This separation of the constituent physical properties h a s not been observed in the other versions of the FRF, or their plots. Fig. 2.11 shows a n example of a plot of this form for a system with a combination of both viscous and structural damping. Fig. shows the Real P a r t which has a slope of a t the axis crossing point, which is itself a t the undamped system natural frequency, Fig.
Fig. 2.11 Inverse FRF plot for system with (a) mixed, and viscous damping
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shows' a straight line whose slope is given by the viscous damping rate, c, and whose intercept at is provided by the structural damping coefficient, d. The potential of this form of presentation, if it carries over to the more general case of MDOF systems, soon becomes apparent.
(iv) The Nyquist or Argand plane plot is widely used and is a very effective way of displaying the important resonance region in some detail.
Fig. 2.12 shows Nyquist-type FRF plots corresponding to the viscously-damped SDOF system previously illustrated in Figs. 2.9 and 2.10. As this style of presentation consists of only a single graph, the missing information (in this case, frequency) must be added by identifying the values of frequency corresponding to particular points on the curve. This is usually done by indicating specific points on the curve a t regular increments of frequency. I n the examples shown, only those frequency points closest to resonance are clearly identifiable because those away from this area are very close together. Indeed, it is this feature
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of distorting the plot so a s to focus on the resonance area-
that makes the Nyquist plot so effective for modal testing applications.I t is clear from the graphs in Fig. 2.12, and also from the companion set i n Fig. 2.13 for hysteretic damping, that each takes the approximate shape of a circle. In fact, as will be shown below, within each set one is a n exact circle (marked by *), while the others only approximate to this shape. For viscous damping, it is the mobility which traces out a n exact circle while for hysteretic damping, it is the receptance and accelerance which do so. I n the other cases, the degree of distortion from a circular locus depends heavily on the amount of damping present becoming negligible a s the damping decreases.
Having shown the most common x-y plots used to present FRF data, it is instructive now to provide a n illustration of the full three-dimensional quantity which the FRF constitutes. An isometric projection of the Re vs. Im vs. Frequency plot of the Receptance FRF of a n SDOF system with viscous damping is shown in Fig. 2.14. From this illustration it is possible to visualise the three projections already shown in (i), and (iv) above a s well a s the original three-dimensional curve itself. I t is not difficult to understand why this type of presentation is not widely used in practice: its interpretation for more complex systems with many
can become extremely difficult!
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Fig. 2.12 Nyquist FRF plots for Fig. 2.13 Nyquist FRF plots for
SDOF system with SDOF system with
viscous damping structural damping
(a) Receptance; (a) Receptance;
(b) Mobility; Mobility;
(c) Accelerance (c) Accelerance
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Fig. 2.14 Three-dimensional plot of SDOF system FRF