Introduction to Acoustics

a)
b)
Fig. 28.2a,b Two ways of representing a vibrational mode
in a Korean pyeongyeong: (a) The shape at maximum
bending; (b) The neutral shape with vectors
timates by a second estimation process. The advantage
of global curve fitting is that more-accurate frequency
and damping estimates can potentially be obtained by
processing all of the measurements rather than relying
on a single measurement. Another advantage is that, because
damping is already known and fixed as a result of
the first step, the modal coefficients are more accurately
estimated during the second step. Both the complex exponential
and rational fraction polynomial methods can
be formulated to obtain global estimates from a set of
measurements [28.6].
28.2.5 Real and Complex Modes
The assumption of proportional viscous damping
implies the existence of real, or normal modes. Mathematically,
this implies that the damping matrix can
be defined as a linear combination of the physical
mass and stiffness matrices. Physically, all the
points in a structure reach their maximum excursion,
in one or the other direction, at the same time. The
imaginary part of the FRF reaches a maximum at reso-
28.3 Mathematical Modal Analysis
In mathematical modal analysis, one attempts to uncouple
the structural equation of motion by means
of some suitable transformation, so that the un-
Modal Analysis 28.3 Mathematical Modal Analysis 1133
µm/s
–500 0 500
Fig. 28.3 Gray-scale representation of a vibration mode in
a violin (courtesy of George Bissinger)
nance, and the Nyquist circle lies along the imaginary
axis.
Some structures exhibit a more complicated form of
damping, and the mode shapes are complex, meaning
the phase angles can have values other than 0 or 180 ◦ .
Different points on the structure reach their maxima at
various times as in a traveling wave pattern. The imaginary
part of the FRF no longer reaches a maximum at
resonance nor is the real part zero. The Nyquist circle is
rotated at an angle in the complex plane. When damping
is light, the proportional damping assumption is generally
an accurate approximation, [28.5] although it can
be argued that a complex-mode formulation is essential
to preserve accuracy in damping evaluation [28.7].
28.2.6 Graphical Representation
One of the nice features of experimental modal testing
is the way that the modes can be represented graphically.
Animations of the vibration can be viewed from
any angle to comprehend complex mode shapes. Static
representations include the shape at maximum bending,
and the neutral shape with vectors, as shown in Fig. 28.2.
Another representation is shown in Fig. 28.3.
coupled equations can be solved. The frequency
response of the structure can then be found by summing
the respective modal responses in accordance
Part H 28.3