Árbók VFÍ/TFÍ - 01.06.2004, Side 213

A sketch of a possible solution to the Duffing equation is shown in
Figure 1 where A0 represents the amplitude of the response due to
the external load Fcos(wt) and fi represents the frequency for the
constants a, c, y, greater than zero. The solid lines on the figure
show the stable part of the solution and the dashed lines the unsta-
ble part. The solution will jump abruptly between the stable solu-
tions at a certain frequency, commonly called the jump frequency.
Hysteresis was experienced in the shallow water sloshing experi-
ments for several configurations of tank layout, water depth and
shaking amplitude. The dependence on initial conditions of the
water response to shaking was investigated by either increasing
the frequency of sloshing from values below the resonant frequen-
cy or decreasing the frequency from values above the resonant fre-
quency.
Although hysteresis is fairly known in some engineering applica-
tions it has not been investigated in detail for sloshing of water.
This is especially true in the case of shallow water sloshing as its
main engineering application, the tuned liquid damper, is a fairly recent field of study
(see e.g. Fujino et al., 1988,1992; Sun et al, 1989; Koh et al., 1994; Gardarsson 1997; Reed
et al., 1998; Gardarsson et al., 2001). Sloshing of deep water has been investigated fairly
extensively (see e.g. Ockendon et al., 1973, 2001; Shemer, 1990; Faltinsen et al., 2000, and
2001) due to its many applications, such as in fuel tanks and ship application. In none of
these cases has the hysteresis phenomena been investigated in detail but it is briefly dis-
cussed in e.g. Lepelletier and Raichlen (1988), and in Faltinsen and Timokha (2002).
In this paper hysteresis in shallow water sloshing will be explored for one particular con-
figuration of a rectangular tank to demonstrate the main characteristics of sloshing of
water close to the resonant frequency.
Experimental setup
The experiments were performed on a precise shaking table at the University of Southern
California earthquake laboratory. Figure 2 depicts the general setup of the experiments.
The shaking table has a 1.2 m by 1.2 m platform, which moves in a single horizontal direc-
tion with a hydraulic system on a platform that is anchored to a concrete floor. The table
was designed and constructed for general structural testing, which usually involves
much greater specimen masses than were used in the present experiments. Hence, over-
loading was not a consideration.
The general experimental setup consisted of a rec-
tangular plexiglas tank mounted on a load cell,
which was bolted directly onto the shaking table.
The length of the tank discussed in this paper was 59
cm, 33.5 cm wide, and 30 cm deep. The tank walls
were made of 1.27 cm thick plexiglas with the bot-
tom being 1.9 cm thick to ensure sufficient rigidity.
The wave response was measured with capacitance-
type wave gauges. The gauge itself is made of a
Tantalum rod (0.5 mm in diameter) with an oxidized
surface, which acts as dielectric for the capacitance-
type wave gauge. The uniform tantalum oxide coat-
ing was achieved by anodizing the Tantalum rod in
weak citric acid solution. The original idea of the use
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