Dr V. Gershkovich
University of Melbourne, Australia
Modal analysis is an engineering discipline which aims to establish structural properties (integrity,
strength, rigidity, damping etc.) on the basis of spectral measurement. Vibration measurement is a convenient
and practical type of measurement, because vibrational characteristics of any structure are closely related
to its structural properties. For example, a good way to determine if a glass goblet has a structural problem
(such as a crack) is to strike it and listen to its vibration response. The essence of Modal Analysis is to
develop techniques such as structural identification, based on vibration measurement and spectral analysis of
vibration.
Vibration of a structural system is determined by the wave equation with an elliptic operator in spatial
variables. Natural frequencies of the system are determined by the spectrum of this operator. Problems in
Modal Analysis are inverse spectral problems related to such an elliptic operator. One has to reconstruct
properties of the structure (mass, stiffness, and damping) on the basis of measurements of the vibration of
this structure. There are several strategies available to produce vibration, to perform the necessary
vibration response measurement, and to finally reconstruct the structural properties from this information.
One of these approaches is to determine several initial eigenvalues and eigenfunctions and then to
reconstruct structural properties of the system.
Historically inverse spectral problems were initiated by spectroscopy. The inverse spectral problem was
formulated in 1882 by Sir A. Schuster, who also created the term "spectroscopy". He formulated the problem as
follows: "Find out the shape of the bell by means of the sound which it is capable of sending out". The
uniqueness of the solution is the problem of the (non) existence of isospectral manifolds.
The developed theory relates to the Laplace operator and to reconstruction of the manifold and the Riemannian
metric on this manifold from the complete spectrum of the Laplacian. The well known isospectral problem is of
great importance to applications from this point of view; the existence of isospectral manifolds suggest
non-uniqueness of the solution to the problem. Practically this means the possibility exists to obtain a
completely erroreous solution to the problem.
The inverse spectral problem which arises in Modal Analysis differs strongly from classical inverse spectral
problems. A feature of Modal Analysis is that due to physical limitations one can reconstruct only a small
number of natural frequencies of the system. However one can also reconstruct the corresponding
eigenfunctions which produce substantial additional information and allows one to reconstruct (under
favorable circumstances) the structural properties of the system.
The proposed scheme allows determination of the necessary number of eigenfunctions to reconstruct structural
properties and to properly plan the conduct of the experiment prior to performing actual measurements. The
problem of "isospectral structures" in this theory relate directly to justification of numerical
algorithms.
This approach is very natural also to problems of crack detection. The simplest version of this problem
relates to perturbation of the coefficients of the equation by a (one) step function with a small support.
The problem is to determine the location of the support on the basis of perturbation of the eigenvalues and
eigenvectors. More complicated problems relate to the presence of multiple and long cracks.
