Civil Engineering Reference
In-Depth Information
Wavelet-based nonstationary
vibration analysis of a
simple nonlinear system
The previous chapters have been devoted to the response analysis of linear
systems subjected to nonstationary random vibrations. In these chapters,
the basic idea of wavelets and the usefulness of wavelet-based analysis have
been explained. The theory of wavelet-based analysis is used to develop
and formulate the system equations of motions and transfer those to the
wavelet domain and solve using wavelet coefficients. However, not all the
structures behave in a way that could be well approximated by a linear
system. In reality, the responses of a structure may deviate far from linear
considerations [34-38]. The response of, say, buildings to earthquakes is
a nonlinear dynamic problem. The readers should keep in mind that at
this point we are talking about the nonlinearity present in the material
model of the structure. However, it should also be borne in mind that we
are not concerned about how to represent material nonlinearity; rather,
we are focused only on how to formulate the equations of motion once
we consider nonlinearity in our assumptions and how we solve the system
dynamics using the wavelet analytical technique. The basic background
theory regarding wavelets as discussed in previous chapters is used in this
chapter too; however, the problem has become more complicated due to
introduction of nonlinearity into the system. The main objective is to pro-
vide an insight into the formulation of the problem related to obtaining the
nonlinear response of the system using the wavelet-based approach.
5.1 NONLINEAR SYSTEM
So far, we have considered linear structural systems in the analysis.
However, most of the engineering structures include some degree of nonlin-
earity under real operating conditions. Though an assumption of a struc-
ture being linear is justified based on dynamic conditions and magnitude
of external input excitation, nonlinear analysis of a structural system may
become necessary as per design requirements that are aimed to improve the
111
