W. Richard Bowen and Nidal Hilal 4

a frequency domain such that the relative magnitude of each frequency
component may be evaluated and the resonance spectrum constructed.
The motion of the cantilever may be modelled as a simple harmonic
oscillator (SHO) with an increased effective mass due to the contribution
of the fluid in close proximity to the beam. Simple approximations
describing the contribution of the fluid mass are described by Oden et al.
(1996) [83], who model the virtual mass of the SHO as the combination
of the cantilever mass and the mass of a spherical fluid volume enveloping
the cantilever. More advanced models are considered by Sader (1998)
[67], who present analytical expressions for the hydrodynamic functions,
which determine the hydrodynamic loading due to the motion of the
mass of fluid around a rectangular beam. The theory considers cantilevers
with rectangular and cylindrical cross-sections for which the length
is far greater than the diameter or width, respectively.
For beams of a circular cross-section, the analytical expression for the
hydrodynamic function � has been established previously; however, rectangular
beams of finite thickness are far more complicated to describe.
Sader (1998) derives a correction factor �(�) which relates the circular
cross-section solution to that of an infinitely thin rectangular cantilever
based on the approximate relationship previously described by Tuck
(1969) [69] such that the hydrodynamic function � rect is given by:
� ( �) � �( �) � ( �)
rect circ
where the correction factor �(�) is of the form
and
9.4 dETERMINATION OF RHEOLOgICAL PROPERTIES FROM RESONANCE SPECTRA 257
�( �) � � ( �) � i�
( �)
r i
� ( �), � ( �) � (Re)
r i f
(9.11)
(9.12)
(9.13)
For small amplitude oscillations, the characteristic Reynolds number
depends upon the cantilever width x such that:
� �
Re �
4�
fluid x2
For an SHO the amplitude A(�) is given by
A0�R
A(
�)
≅
⎡
⎢ 2 2 � �
⎢(
� �R)
⎣
Q
2
� � 2
2
1
2 2 2
R
⎤
⎥
⎦
(9.14)
(9.15)