Identification of the dynamic characteristics of nonlinear structures

1 6 . 6 5
1 5 . 7 0
1 6 . 6 5
2
F r e q .
2
IFig.2.3 Real and Imaginary Parts of Reciprocal FRF Data with Stiffness Nonlinearity
R e a l P a r t
2 3 . 4 7
2 7 . 5 5
F r m q .
 2
I m a g i n a r y P a r t .
2 3 . 4 7
.
F r m q .
 2
Real and Imaginary Parts of Reciprocal FRF Data with Damping Nonlinearity

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2 Identification of Nonlinearity Using First-order 
26
So far, it has been demonstrated that both Bode and reciprocal receptance plots can be
used to detect the existence of nonlinearity, and that the latter technique can tell whether
the nonlinearity is of stiffness or damping type in the case where the measured modes are
effectively real. However, these methods can only be used to provide rough and basic
demonstration of the existence of nonlinearity. It is not possible to quantify the extent of
the nonlinearity based on these methods. In what follows, an alternative method of
nonlinearity detection the isometric damping plot technique will also be discussed.
2.3.2 
ISOMETRIC DAMPING PLOT TECHNIQUE
It has been established 
that structural nonlinearity can be detected by inspection of the
isometric damping plots which can be calculated from measured FRF data. The argument
which supports this technique is generally believed to be that structural nonlinearity
(usually stiffness nonlinearity) distorts the spacing of frequency response data around the
Nyquist ‘circle’ from their positions when no nonlinearity exists. Since the distortion
caused by nonlinearity is systematic, the consequent distortion of the damping estimate
plot using different pairs of points around the Nyquist circle will display a specific pattern
depending on the type of nonlinearity. These patterns in the damping plot can then be
and compared to detect and possibly to identify the nonlinearity. The
mathematical basis of this technique will be discussed next with the new explanation for
the reason why damping estimates vary when different pairs of frequency points are
used.
Again, suppose that the residual effect of other modes can be neglected and that the modal
constant is effectively real for the mode to be analysed, then the receptance 
around
mode can be expressed as that of equation (2-l). When 
is plotted in the
Nyquist plane, a circle as shown in Fig.2.5 can be obtained. If the data are measured on a
linear structure, then the damping loss factor of 
mode can be calculated as follows:

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2 Identification of Nonlinearity Using First-order 
2 7
Fig.2.5 Nyquist Circle of Receptance Data
r r
Adding equations (2-3) and 
the damping loss factor is given by:
(tg 
+ 
(2-4)
When different combinations of points 
are used, a flat plane which is the surface
plot of the estimated damping ratio 
against its two variables 
and 
can be
obtained in the case of linear FRF data.
On the other hand, if the measured FRF data are from a nonlinear structure, distortion of
the isometric damping plot (no longer a flat plane) calculated based on equation (2-5) will,
in general, be expected as shown in Fig.2.6 for the data measured from a Beam/Absorber
structure shown in figure 2.1.

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2
Identification of Nonlinearity Using First-order 
2 7
Re
Fig.2.5 Nyquist Circle of Receptance Data
Adding equations (2-3) and 
the damping loss factor is given by:
(tg 
+