Identification of the dynamic characteristics of nonlinear structures

2 Identification of Nonlinearity Using First-order 
32
C o e f f i c i e n t s
Natural 
Fig.2.8 Analysis Results of Complex Mode Using Inverse Receptance Method
2.3.4
COMMENTS ON CURRENT NONLINEAR MODAL
ANALYSIS METHODS
so far, some of the currently available nonlinear modal analysis methods (based on the
extension of linear modal analysis) have been reviewed. Based on the systematically
abnormal behaviour of measured nonlinear FRF data, qualitative methods as discussed
above can be used to detect the existence of nonlinearity. By comparing FRF data
measured at different input force levels, the overlaid Bode plots can be used to check
whether or not the measured data display nonlinearity. On the other hand, when FRF data
are presented in their reciprocal form, the effect of stiffness and damping nonlinearities
can be separated into the real and imaginary parts of the data and, by examining the plots
of both parts, the type of existing nonlinearity (stiffness or damping) can be revealed.
Also, the isometric damping plot technique can be used to detect the existence of
nonlinearity by investigating the variation of damping ratios calculated using different
pairs of frequency points on the Nyquist circle. The reason for this damping variation in
the case of stiffness nonlinearity, as demonstrated, is due to the different response
amplitudes, and so different effective natural frequencies, of different receptance data
points.
With these methods available, the task of detecting the existence of nonlinearity can be
accomplished reasonably successfully if the structural nonlinearity has some contribution
to the measured first-order 
However, since all these methods are qualitative in

---

2
Identification of Nonlinearity Using First-order 
3 3
nature, it is difficult for them to establish the extent of nonlinearity which a structure
possesses.
The Inverse Receptance method seeks to quantify nonlinearity by establishing the
relationship between modal parameters and response amplitudes: 
and 
The method was developed based on an assumption that the modal constant of the
mode to be analysed is real and constant. Although valid for FRF data measured from
nonlinear SDOF systems, for those measured on practical nonlinear structures, this
assumption is, in general, no longer valid for following reasons:
(i) measured data may contain mode complexity;
(ii) the modal constant of a mode is, in theory, a function of response amplitude.
The effect of mode complexity on the analysis results based on Inverse Receptance
method has been demonstrated and detailed discussions on the existence of genuine
complex modes will be presented later on. Here, only the second point (the modal
constant of a nonlinear system is a function of response amplitude) will be illustrated
based on a 2DOF system with cubic stiffness nonlinearity as shown in Fig.2.9.
Assuming 
is nonlinear and can be expressed as 
(where is
the vibration amplitude of mass 
when it vibrates sinusoidally) and solving the
eigenvalue problem of this system, for the first mode, the natural frequency 
and
modal constant 
of 
can be expressed as:
= 
= 
= 1.0000 kg
4ooooo 
+ 
Fig.2.9 A 2DOF Nonlinear System
=
4 + 
4 + 
2
(2-18)

---