2.2. PART 2: GUIDELINES FOR SYSTEMATIC MODEL DEVELOPMENT 23
as an ersatz fluid with properties in the general range of liquid oxygen (LOX); the effective
bulk modulus of water at room temperature, accounting for radial flexibility of the feedline
structure is B 1:6 10
5
psi, and its mass density is 1:36 10
5
lb-sec
2
/in
4
. On the basis
of continuum theory, the quarter wavelength .L=4/ for the fluid within the feedline is:
e. Dilational fluid deformation: L=4 D
p
B==.4f
/ 207 in.
Clearly, the cross-sectional dimension of the feedline is an order of magnitude less than
L=4. erefore, a fluid model corresponding to a structural “rod” enclosed within the feedline
structure, yet permitted to slide with respect to the rod, is most appropriate.
Since the contained fluid is constrained to move laterally with the feedline structure, the
“lateral” mass per unit length, A :0213 lb-sec
2
/in
2
, is the sum of structural and fluid com-
ponents. erefore, employing the relevant technical (Euler–Bernoulli) beam theory, the lateral
L=4 estimate for the feedline is
f. Feedline bending deformation: L=4 D
.
=2
/
.
EI=A
/
1=4
=
p
4f
80:8 in.
is result indicates that the ratio of quarter wavelength to feedline diameter is roughly 5
at f
D 50 Hz, suggesting that a feedline structural model accounting for cross-sectional shear
deformation (Timoshenko beam theory) may be most appropriate. Fortunately, most modern
FEM codes employ beam elements that are based on Timoshenko beam theory.
2.2.12 MODAL DENSITY AND THE EFFECTIVENESS OF FINITE
ELEMENT MODELS
Finite element modeling is an effective approach for study of structural and mechanical system
dynamics as long as individual vibration modes have “sufficient frequency spacing” or “low modal
density.” Modal density is typically described as the number of modes within a 1/3 octave fre-
quency band (f
0
< f < 1:26f
0
). When modal density of a structural component or structural
assembly is greater than 10 modes per 1/3 octave band, details of individual vibration modes are
not of significance and statistical vibration response characteristics are of primary importance.
In such a situation, the Statistical Energy Analysis (SEA) method [18] applies.
Table 2.3 [15] gives formulae for modal density (as a mathematical derivative, d n=d!
(n D number of modes, ! D frequency in radians/sec), for typical structural components.
e above modal density relationships relate to structural components, rather than struc-
tural assemblies that are composed of a variety of components. More reliable estimation of a
particular system’s modal density is based on the assembled system’s modes.
2.2.13 ILLUSTRATIVE EXAMPLE: FLUID-FILLED CIRCULAR
CYLINDRICAL SHELL
e following example system, consisting of a fluid-filled, thin-walled circular cylindrical shell
(see Figure 2.6), was extensively studied by Abramson [19] during the mid-1960s. It contin-
ues to offer excellent, comprehensive guidance for modeling of launch vehicle propellant tank
24 2. DEFINITION OF TEST ARTICLE FINITE ELEMENT MODELS
Table 2.3: Modal densities of typical structural components
Component Motion Modal Density, dn/dω Additional Data
String Lateral
L/(π T/ρA)
T = tension, A = area
ρ = mass density
L = length
Rod Axial
L/(π E/ρ)
E = elastic modulus
Rod Torsion
L/(π G/ρ)
G = shear modulus
Beam Bending
L/(2π)(ω EI/ρA)
-1/2
EI = fl exural stiff ness
Membrane Lateral
A
s
ω/(2π)(N/ρh)
N = stress resultant
A
s
= surface area
Plate Bending
A
s
/(4π) D/ρh)
D = plate fl exural stiff ness
h = plate thickness
Acoustic Dilational
V
o
ω
2
/(2π
2
)( B/ρ)
3
B = bulk modulus
V
o
= enclosed volume
structures. e geometry, properties, and boundary conditions for the present example are as
follows.
• Geometry: diameter (D D 300 in), length (L D 600 in), wall thickness (h D 1 in).
• Material: aluminum (E D 10
7
psi), Poisson’s ratio ( D 0:3), density (
s
g D 0:1 lb/in
3
).
• Fluid: water (
f
g D 62:4 lb/ft
3
D :036 lb/in
3
).
• Fluid B.Cs: fluid free surface at Z D 600 in, blocked tank bottom at Z D 0 in.
• Structure B.Cs: bottom (Z) axial restraint, and free to radially expand without bending
restraint.
• Ullage Pressure: P
0
D 0 psi, or 30 psi.
e structural shell mode shapes associated with empty and fluid-filled shell are of the
form:
ˆ.z; Im; n/ D cos.mz=2L/ cos.n/ or cos.mz=2L/ sin.n/;
m D 1; 3; 5; : : :; n D 0; 1; 2; 3; : : :: (2.51)
Closed-form expressions for empty structure natural frequencies (without and with ullage
pressure) are provided below. ey are associated with bulge (n D 0), lateral (n 1) and shell
breathing (n > 1) for the empty shell are
f
mn
D
1
2
k
1
C k
2
C k
3
s
h
1=2
; (2.52)
2.2. PART 2: GUIDELINES FOR SYSTEMATIC MODEL DEVELOPMENT 25
Z
Y
L
X
Fluid-Free Surface
The tank bottom is
axially fixed and free
to radially expand
D
Figure 2.6: Fluid-filled circular cylindrical shell.
where
k
1
D
Eh
R
2
m
L
4
m
L
2
C
n
R
2
2
(membrane stiffness parameter) (2.53)
k
2
D
Eh
3
12
.
1
2
/
m
L
2
C
n
R
2
2
(flexural stiffness parameter) (2.54)
k
3
D
.
P
o
R
/
m
L
2
C
n
R
2
(ullage “differential” stiffness parameter): (2.55)
In addition, the empty structure’s axial and torsion modal frequencies are (noting that
G D E=.1 C 2//,
f
Z;m
D
1
2
m
L
p
E=
s
; f
;m
D
1
2
m
L
p
G=
s
; (2.56)
respectively.
Closed-form expressions for the fluid-filled structure natural frequencies (without and
with ullage pressure) are presented by Abramson in [19], for the “curious” reader. However, the
intent of this illustrative example is to provide insight into parameters affecting the system’s
modal frequencies.
26 2. DEFINITION OF TEST ARTICLE FINITE ELEMENT MODELS
e first informative result of closed-form modal analysis, provided in Figure 2.7, indicates
the roles of membrane, flexural, and ullage pressure strain energies on empty shell natural (m D
1) frequencies.
Sensitivity of Empty Shell “m = 1” Frequencies to Membrane, Flexure,
and Ullage Pressure Strain Energies
Membrane
Membrane+Flexure
Membrane_Flexure+Ullage
10
2
10
1
10
0
Frequency (Hz)
0 2 4 6 8 10 12 14 16
Circumferential Harmonic (n)
Figure 2.7: Sensitivity of empty shell natural frequencies to strain energy contributions.
e second informative result of closed-form modal analysis, provided in Figure 2.8, in-
dicates the roles of membrane, flexural, and ullage pressure strain energies on fluid-filled shell
natural .m D 1/ frequencies.
e profound effect of fluid mass on shell natural frequencies is summarized in Figure 2.9
for the “no-ullage pressure,” m D 1 modes.
e high modal density and ullage pressure sensitivity of empty shell natural frequencies
are illustrated in Figure 2.10.
Table 2.4 details the numerical natural frequencies corresponding to Figure 2.13. Note
that (1) modal frequencies exceeding 85 Hz are shaded in gray and (2) axial and torsion natural
frequencies are indicated.
e corresponding high modal density and ullage pressure sensitivity of fluid-filled shell
natural frequencies are illustrated in Figure 2.11.
Table 2.5 details the numerical natural frequencies corresponding to Figure 2.11. Note
that (1) modal frequencies exceeding 85 Hz are shaded in gray and (2) axial and torsion natural
frequencies are indicated.
e following insights are gained from the previous results:
2.2. PART 2: GUIDELINES FOR SYSTEMATIC MODEL DEVELOPMENT 27
Sensitivity of Fluid-Filled Shell “m = 1” Frequencies to Membrane, Flexure,
and Ullage Pressure Strain Energies
Membrane
Membrane+Flexure
Membrane_Flexure+Ullage
10
2
10
1
10
-1
10
0
Frequency (Hz)
0 2 4 6 8 10 12 14 16
Circumferential Harmonic (n)
Figure 2.8: Sensitivity of fluid-filled shell natural frequencies to strain energy contributions.
Sensitivity of Shell “m = 1” Frequencies to Fluid-Mass (Ullage Pressure = 0 psi)
Empty
Fluid-Filled
10
2
10
1
10
0
Frequency (Hz)
0 2 4 6 8 10 12 14 16
Circumferential Harmonic (n)
Figure 2.9: Sensitivity of natural frequencies to fluid mass loading.
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