EFFECTS OF MASS OF SAND ON THE DYNAMIC
CHARACTERISTICS OF A THIN WALL STRUCTURE
Danh Tmnxuan
Mechanical
Engineering
Department
Victoria University of Technolow
P.O.Box 14428, MCMC, Vie 8001
Australia
ABSTRACT. The dynamic characteristics of a
thin wall rectangular paraIIelepiped
steel tank
containing sand was studied by experimentaI
modal analysis for varying amount of sand. It
was found that below a certain amount of sand
the natural frequency corresponding to a
particular mode shape increases as the mass of
sand is increased but as the mass of sand is
increased further from that amount the
natural
frequency
would increase. T h e
damping ratio corresponds to a mode shape
always increases with the mass of sand. The
mode shapes themselves may be wiped out due
to increased damping ratio, they may emerge
from existing modes as the result of heavy
coupling due to high damping ratio or they
may be entirely new modes due to the
contribution of the varying body of mass of
sand.
These effects of mass of sand on the dynamic
characteristics of the thin wall structure shoud
be taken into account in evaluating dynamic
response of the structure.
NOMENCLATURE
f
s
: frequency (Hz)
: damping ratio (%)
1. Introduction
Thin w&I structures are often used to store
and transport bulk materials. These structures
are often subjected to random vibrations, for
example due to land and sea transportation
loads or earthquakes. A knowledge of their
dynamic characteristics is essential
for
predicting their dynamic response.
It has weII been recognised
in the case of thin
waII structures containing Iiquid that the fluidstructure interaction effect between the liquid
and the wall flexibility must be taken into
account to reflect the dynamic behaviour of the
structure [l-3]. Thin walI structures containing
bulk granular material have not received as
much attention
In this study the dynamic characteristics of a
rectangular steel tank
containing sand is
studied by experimental modal analysis.
2. The structure
The structure is an open rectangular tank of
dimensions shown in Figure 1. Due to its
continuous nature and relatively flexibility the
empty tank exhibits a great number of normal
modes even in low frequency range. From the
view point of structural system, when the
empty tank is loaded with sand, even by a
small amount it is changed to a different
system consisting of two materials of very
different behaviour and constitutive
equations.
Whenever the amount of sand is changed, the
tank-sand structure is changed to a different
structure. However, in this study the tank is
considered as the basic structure and the
effects of increasing amount of sand on the
dynamic characteristics of the tank structure
is studied.
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The effects of changing mass of bulk material
on the dynamic response of the system has not
received much attention. In studying the
sloshing of fluid in containers Barron and
Chng [4] reported changes in predominant
sloshing frequency for cases of an empty
cylindrical tank and with l/4, I/2 and 314
full of water. In another study 5 the author has
investigated the cause of thin wall structure
containing bulk granular material , it was
shown that the interaction between bulk
granular material and thin walls of the
structure influence both the stiffness and the
damping characteristics of the solid structure
It has been recognised
that when a bulk
granular material is contained in a solid
structure it exhibits many features very
different those of a liquid or a soil mass [6-7).
m e c h a n i c a l behaviour
is strongly
its
influenced by moisture and compacting
pressure. Therefore the interaction between
bulk granular material and the thin walled
structure is affected by many parameters:
moisture, confining pressure, coeffkient
of wall
friction, damping mechanism, effect of bulk
the
mass a n d
stiffness
material on
distribution.
3. experimental procedures and analysis
The tank is freely suspended at four corners by
four soft springs of stiffness of 1200 N/m. The
mass of the structure is 11.34 kg. Dry sand of
was loaded into the tank in steps of 2 kg from
0 kg up to 30 kg.
The dynamic characteristics of the structure
were determined by experimental modal
analysis techniques using the impact hammer
test. The tank is modelled by 135 grid points as
shown in Figure 2. The tank was first tested
empty, then with increasing mass of sand. At
e a c h l o a d i n g s t e p the FRF (Frequency
Response Function) curves at all grid points
were recorded on a frequency analyser (B & K
Dual Channel Signal Analyser
2032) and
transferred to a modal analysis software
(Structural Measurement Systems SMS Modal
3.0) for later analysis, the coherence curves
and signal noise ratio curves were also
monitored for each reading to ensure the
quality of the FRF curves. A typical set of FRF
curve for a grid point with various masses of
sand is shown in Figure 3, where the frequency
range used is 0 - 200 Hz. As the damping of
granular material is nonlinear, care was taken
to insure that the impact caused by the
hammer hitting on the walls did not introduce
irreversibility in the system. This was effected
by preliminary tapping all around the structure
once the masts of sand had been added and by
controlling the intensity of striking to be
uniform and by consistent striking on the part
of the operator. Absence of irreversibility
during the course of the experiment was
checked and confirmed by comparing the FRF
curves obtained at the same grid point at the
beginning and at the end of the experiment. It
w a s found that the FRF at grid points were not
affected by the striking of the hammer.
As expected, it was observed that the thin wall
structure exhibited a great number of normal
m o d e s , only a number of prominent modes in
the range of O-110 Hz were analysed. A mode
shape was observed in isometric view as well
as in plan, elevation and end views ( viewing
along z, x, y respectively). Results of the
analysis is presented in Table I showing the
variation of natural frequency f in Hz and
damping ratio 5 (%) with mass of sand in the
container and a number of mode shapes are
shown in Figure 4.
4. Rcsulte.
It was found that as the mass of sand is
increased the frequency and damping ratio of a
normal mode changes and even mode shapes
of the container are changed. The scheme of
naming mode shapes is as follows: they are
named by a single alphabet name only if the
m o d e shape is not changed to a neighbouring
mode shape (eg. mode A, mode C) and are
named by an alphabet followed by an index if
they are changed to neighbouring modes as
the mass of sand increases (eg. mode BI,
D3,...).
a) the change in frequency and damping
mode shape:
ratio for a particular
For small amount of sand, up to 8 kg for the
container
studied,
the frequency for a
particular mode shape decreases with the mass
of sand while the damping ratio increases with
the mass of sand as shown by mode shape A,
Bl, B2, C, Dl, D2. The change of frequency
and damping ratio can be quite large, for
example for mode shape A , at 8 kg of added
sand the frequency changes from 48.4 Hz to
38.9 Hz ( 20%) and damping ratio increases
from 0.9% to 6.8%.
For larger amount of sand ( greater than 8 kg),
while the damping ratio increases with the
mass of sand as previously, the frequency now
increases with mass of sand as shown by
mode B3, ,B5, D5, D7.
b) the change in mode shapes:
As the mass of sand increases, more modes are
observed for the same frequency range as
shown in Figure 3 but some modes may be
.--
wiped out. New modes formed may be only
slightly different or entirely different from the
previous mode, for example mode El is formed
at 2 kg of sand and mode C formed at 6 kg of
sand are entirely new mode while the change
from mode shape Dl to mode D2, for example,
is gradual.
As the damping ratio of a particular mode
increases, it tends to couple heavily with
neighbouring modes which could result in it
being effectively wiped out as is the case of
mode A and C which no longer exist as mass
of sand is greater than 8 kg; or results in
emergence of a new mode which differs only
slightly from the previous mode (as in the
case of mode Bl to B5, Dl to D7 ).
For the case of mode Bl and C, it should be
noted that they emerge only when the maw of
sand is 2 kg and 4 kg respectively, it is not
observed when the tank is empty indicating
that as the mass of sand is changed the tank is
effectively coupled with another sub-structure:
the sand which is a changing continuous solid
would contribute a great number of modes to
the tank-bulk material system which would
exhibit a number of new modes, some could be
entirely different from the modes of the present
system or are emerging from modes of the
present system.
5. Conclusions.
The effects of mass of sand on the dynamic
characteristics of the thin wall carrier can be
summarised
as follows:
a. when the mass of sand is small, the
frequency of a particular mode shape decreases
with mass of sand indicating that the added
mass effect due to the bulk material is more
pronounced than the effect of added stiffness
to the structure,
b. when the
frequency of a
with mass of
stiffness effect
mass of sand is large the
particular mode shape increases
sand indicating that the added
is more pronounced,
c. the damping coefficient of a particular mode
shape always increases with mass of sand,
d. some modes may be wiped out due to
increased damping ratio and heavy modal
coupling,
e. entirely new modes may be formed due to
the changing contribution of the continuous
solid bulk material substructure,
f. increased modal coupling due to increased
damping and the changing contribution of the
bulk material induce gradual emergence of
mode shapes sharing some features of previous
modes.
These effects should be taken into account in
predicting the dynamic response and the
fatigue strength of thin wall structures.
REFERENCES
[l] Gupta, RK. and Hutchinson G.L.
Effects of wall flexibility on the dynamic
response of liquid storage tank
Eng. Struct., 13, pp 253-267.1991.
[Z] Holi, Y. et al
Coupling vibration analysis of fluid and
structure using an FEM Displacement method,
Proc of Asia-Pacific Vibration Conference ‘93,
1, pp 126-131, Kitakyushu, November 1993.
[3] Eguchi, Y. et al
Vibration of a thin cylindrical shell induced by
fluid overflow
Proc of Asia-Pacific Vibration Conference 93, 1,
pp 114-l 19, Kitakyushu, November 1993
[4] Barron, R. and Chng, S.W.
Dynamic analysis and measurement of sloshing
offuid in containers
Journal of Dynamic Systems, Measurements
and Control, 111, pp 83-90, 1989
[5] Tra-n, D and He, J.
Effects of bulk materials on this dynamic
characteristics of thin shell carrier.
Proc of Asia-Pacific Vibration Conference 93, 4,
pp 1551-1556, KitaKyushu,
November 1993
[6] Gorenc, B.E.
Guidelines for the acssessment of Loads on
Bulk Solids Containers
IEAUST, pp56-59,1986
[7j Stepanoff. A.J.
Gravity Flow of Bulk Solids And Transportation
of Solids in Suspension
John Wiley Bs Sons, New York, pp 35-47, 1969
959
960
(all
dimensions
in
mm)
Figure 2 :
Grid Coordinate System
Figure 1 :
Thin wall steel tank
(1.5mm thick)
Figure 3 : FRF curves at a point for various masses of sand
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(b) Mode Dl
(a) Mode A
(e) Mode D3
(c) Mode D2
Figure 4.
Some Mode shapes of the tank
shown in Table I
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