Excitation and response 53
modes for a certain wave type we therefore have to rely on various approximate methods.
Finite element methods (FEM, BEM) implemented in advanced computer programs, such
as e.g. ANSYS™, ABAQUS™ and FEMLAB™, have made such methods powerful
tools. At the same time one has the ability to use advanced experimental methods, modal
testing or modal analysis, to compare and to give feedback on the calculated results. In
an interactive way, one may then improve on the mathematical model of the structure
under investigation.
Another important analytical tool is statistical energy analysis (SEA). We shall
therefore give an overview of this method later in Chapter 7. As distinct from the finite
element methods, SEA is what we might call an energy flow method. It will not give any
detailed description of the oscillating motion at a given frequency but gives us a picture
of the energy flow in the system averaged over wide frequency bands. The fundamental
basis of the method is that each element or subsystem making up the structure under
investigation exhibits a number of natural frequencies inside these frequency bands. The
calculated response to an excitation is therefore always some mean value for one or more
frequency bands. In a vast number of cases, not only in building acoustics but also in
general noise problems, this is sufficient information. Several commercial computer
program packages are available, e.g. AutoSEA™, SEADS™ and SEAM™.
In general, we know that the response to an input to a mechanical system such as a
plate, a beam or a shell will be dependent on position. Detecting a resonance when the
driving frequency coincide with one of the natural frequencies then presupposes that the
amplitude of the vibration mode, associated with this natural frequency, is different from
zero in the driving point. In a measurement situation for a proper mapping of natural
frequencies several driving input points have to be used.
An experimental modal analysis does not map the natural frequencies of a structure
only but determines the natural vibration patterns, the modal shape of the structure.
Putting it simply, determining the modal shape is based on the measurement of a number
of transfer functions for the structure. A force is applied in one or more points and for
each driving point the response is measured at a number of positions distributed over the
whole structure. From the measured resonance frequencies, one may estimate the natural
frequencies. Simultaneously, one has at each of these frequencies and at each measuring
point the information on how the structure vibrate both in amplitude and phase, i.e. one
has an estimate of the associated modal shape. From this information it is possible to
construct a model of the structure for solving the inverse problem: calculating the
response to an arbitrary excitation. This is possible due to the response being a
combination of the responses of the separate modes. An introduction to this technique
can be found in the book by Ewins (1988). Modal analysis is, as mentioned above, an
important measurement method giving feedback to finite element methods.
2.6 References
ISO 2041: 1990, Vibration and shock – Vocabulary.
Bendat, J. S. and Piersol, A. G. (2000) Random data: Analysis and measurement
procedures, 3rd edn. John Wiley & Sons, New York.
Blevins, R. D. (1979) Formulas for natural frequency and mode shape. Van Nostrand
Reinhold Company, New York.
Ewins, D. J. (1988) Modal testing: Theory and practice. Research Studies Press Ltd,
Taunton; John Wiley & Sons, New York.