22 2. DEFINITION OF TEST ARTICLE FINITE ELEMENT MODELS
Table 2.2: Quarter wavelength relationships for typical structural components
Basis Component Wave Type L/4 Additional Data
Continuum
Mechanics
3-D Elastic
Dilational
( E/ρ)/(4f *)
E = stretch modulus
G = shear modulus
B = bulk modulus
ρ = mass density
Shear
( G/ρ)/(4f*)
3-D Acoustic Dilational
( B/ρ)/(4f *)
Technical
eory
String Lateral
( T/ρA)/(4f *)
T = tension
A = cross-sectional area
EI = fl exural stiff ness
Rod
Axial
( E/ρ)/(4f*)
Torsion
( G/ρ)/(4f *)
Beam Bending
(π/2)(EI/ρA)
1/4
/ 2π f *
Membrane Axial
( N/ρh)/(4f *)
h = plate thickness
D = plate fl exural stiff ness
N = in-plane stress resultant
Plate Bending
(π/2)(D/ρh)
1/4
/ 2π f *
G 3:84 10
6
psi, 2:59 10
4
lb-sec
2
/in
4
), the following quarter wavelengths .L=4/
are defined for continuum mechanics theory:
a. Dilational deformation: L=4 D
p
E==.4f
/ 982 in.
b. Shear deformation: L=4 D
p
G==.4f
/ 609 in.
Clearly, the cross-sectional dimension of rod, beam, and plate-shell components of the
largest feasible launch vehicle and spacecraft structures are substantially smaller than the above
L=4 estimates. erefore, models based on technical theories are most feasible.
2.2.11 ILLUSTRATIVE EXAMPLE: ALUMINUM LAUNCH VEHICLE
FEEDLINE
Consider a large-scale launch vehicle’s propellant feedline with the following cross-sectional
dimensions (not representative of any particular system):
Outer diameter: OD D 15 in; wall thickness: h D 0:5 in:
e feedline’s cross-sectional parameters are A D 22:7 in
2
, I D 599 in
4
. erefore, employing
the relevant technical theory formulae, the L=4 estimates are:
c. Axial deformation: L=4 D
p
E==.4f
/ 982 in.
d. Torsional deformation: L=4 D
p
G==.4f
/ 609 in.
In order to properly consider lateral bending dynamics of the propellant feedline, the
contained fluid must be considered. For the purposes of the present discussion, water is employed
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