40 3. SYSTEMATIC MODAL TEST PLANNING
Z
Z
Y
Y
X
X
Mode 1: 18.67 Hz (KE
X
= 94%) Mode 2: 19.31 Hz (KE
Y
= 89%)
Figure 3.3: Mode 1: 18.67 Hz (KE
x
D 94%), Mode 2: 19.31 Hz (KE
y
D 89%).
Table 3.1: P5 test article FEM modal kinetic energy distributions
X Y Z X Y ZTruss Grapple
Component KE (%)Directional KE (%)
Fixture
Mode Freq (Hz)
Truss Grapple
Component KE (%)Directional KE (%)
Fixture
Mode Freq (Hz)
3.1.3 ILLUSTRATIVE EXAMPLE: AXISYMMETRIC SHELL FINITE
ELEMENT MODEL
e shell structure, shown in Figure 3.4, consists of five substructures, namely: (1) a lower cylin-
drical skirt (fully fixed at its base), (2) a lower hemispherical bulkhead, (3) lower cylindrical
section, (4) upper cylindrical section, and (5) upper hemispherical bulkhead.
In addition, allocation of unassembled component mass and stiffness matrices within the
5616 DOF model are illustrated in Figure 3.5.
e overall dimensions of the aluminum structure are length, L D 100 in, radius, R D 20
in, and wall thickness, h D 0:4 in. It should be noted that this illustrative example structure
does not represent a realistic design. e rather high thickness-to-radius ratio, h=R D 1=50, was
3.1. PART 1: UNDERSTANDING MODAL DYNAMIC CHARACTERISTICS 41
Z
Y
X
•
•
•
System Point
Substructure Point
Substructure Boundary Point
Figure 3.4: Illustrative example shell structure.
Skirt Lower Dome Upper DomeLower Cylinder Upper Cylinder
0
1000
2000
3000
4000
5000
0
1000
2000
3000
4000
5000
0
1000
2000
3000
4000
5000
0
1000
2000
3000
4000
5000
0
1000
2000
3000
4000
5000
0 2000 4000 0 2000 4000 0 2000 4000 0 2000 4000 0 2000 4000
Figure 3.5: Allocation of shell structure component matrices.
selected to produce less shell breathing modes in the base frequency band (f < 2000 Hz) than
typical aerospace systems, while including modes of sufficient complexity to illustrate key aspects
of quantitative normal mode metrics. e subject structure, fully constrained at the bottom of
the lower cylindrical skirt, has 150 modes in the 0–1453 Hz frequency band. If the example
structure is viewed as a 1/20th scale model, the full-scale set of 150 modal frequencies becomes
0–73 Hz, which is representative of the range of interest for some spacecraft applications.
Modal kinetic and strain energy grouped sums, partitioned by subsystem component, yield
the distribution of component activity for a particular mode. It should be noted that the com-
ponent kinetic and strain energies are not necessarily distributed in the same manner as one
another when they are computed on the basis of separate component mass and stiffness matri-
42 3. SYSTEMATIC MODAL TEST PLANNING
ces in accordance with
f
KE
m
g
n
D
f
Œ
M
m
f
ˆ
g
n
g
˝
f
ˆ
g
n
; KE
mn
D
X
DOF
f
KE
m
g
n
1
(modal KE for component m, mode n)
f
SE
m
g
n
D
f
Œ
K
m
f
ˆ
g
n
g
˝
f
ˆ
g
n
=
n
; SE
mn
D
X
DOF
f
SE
m
g
n
1
(modal SE for component m, mode n):
(3.6)
Modal dynamics of axisymmetric shell structures are composed of “body” modes (i.e., ax-
ial, lateral two orthogonal directions, torsional, and bulge) and “breathing” modes associated
with n > 1 circumferential harmonics as well as axial harmonics). In order to systematically
classify modes of geometrically (but not necessarily physically) axisymmetric shell structures,
consider the seven “body” geometric patterns for successive shell circumferential stations illus-
trated in Figure 3.6.
e seven patterns are associated with cross-sectional lateral (“TX” and “TY”), and axial
(“TZ”) translations, pitch, yaw, and torsional (“RX,”“RY,” and “RZ”) rotations, and radial bulge
(“TR”) translation. In “circumferential harmonic terms,” the above seven patterns represent n=0
and 1 motions (or load patterns).
By organizing the above-described geometric patterns as a body displacement transfor-
mation matrix, Œ
b
, the discrete FEM shell displacements, Œˆ, are expressed as
Œ
ˆ
D
Œ
ˆ
b
C
Œ
ˆ
r
D
Œ
b
Œ
'
b
C
Œ
ˆ
r
; (3.7)
where Œˆ
r
represents remaining (residual) displacements that are not represented by the body
patterns.
Employing weighted linear least-squares analysis, as described below, each system normal
mode may be partitioned into “body” and (remainder) “breathing” components, as follows.
Step 1: Weighted least squares
‰
T
b
Mˆ
D
‰
T
b
M ‰
b
Œ
'
b
C
‰
T
b
Mˆ
r
: (3.8)
Step 2: Orthogonal constraint
T
b
Mˆ
r
D
Œ
0
: (3.9)
Step 3: Generalized “body” modes
Œ
'
b
D
‰
T
b
M ‰
b
1
‰
T
b
Mˆ
: (3.10)
Step 4: “Body” and “breathing” distributions
Œ
KE
b
D
Œ
Mˆ
b
˝
Œ
ˆ
b
;
Œ
KE
r
D
Œ
Mˆ
r
˝
Œ
ˆ
r
: (3.11)
3.1. PART 1: UNDERSTANDING MODAL DYNAMIC CHARACTERISTICS 43
Typical “FX” Load Patch Typical “FY” Load Patch Typical “FZ” Load Patch
Typical “MX” Load Patch Typical “MY” Load Patch Typical “MZ” Load Patch
20
20
10
10
0
0
-10
-10
-20
20
20
10
10
0
0
-10
-10
-20
20
10
0
-10
-20
20
10
0
-10
-20
20
2010
10
0
0
-10
-10
-20 -20
-20
20
10
0
-10
-20
20
10
0
-10
-20
-20
20
20
20
20
0
5
10
10
10
10
0
0
0
0
-10
-10
-10
-10
-20 -20
-20-20
Z
Z
Z
0
5
-5
Z
0
5
-5
Z
Z
Z
Y X Y X Y X
Y X Y X
Typical “FR” Load Patch
Y X
Y X
Figure 3.6: Shell axial station “body” geometric deformation patterns.
Step 5: “Body” and “breathing” total kinetic energies formed by column sums of Equa-
tion (3.11):
Body kinetic energy for mode “n” D KE
b;n
D
X
f
KE
b
g
n
Breathing kinetic energy for mode “n” D KE
r;n
D
X
f
KE
r
g
n
:
(3.12)
e sum of “body” and “breathing” modal kinetic energies for each mode is always unity
(or 100%). In the case of an ideal axisymmetric structure, the “body” and “breathing” modal
kinetic energy distributions are either 0 or 100%. When imperfections and/or localized features
are present (defining a perturbed shell structure) designations of “body” and “breathing” modal
kinetic energy distributions are less distinct. In such situations, the modal kinetic energy desig-
nation of mode “type” may be defined on the basis of majority components (i.e., a “body” mode
has more than 50% body kinetic energy).
44 3. SYSTEMATIC MODAL TEST PLANNING
A yet more subtle description of “body” and “breathing” modal kinetic energies is realized
by analysis of the “body” contribution in Equation (3.7), specifically since
Œ
ˆ
b
D
Œ
‰
b
Œ
'
b
; (3.13)
ˆ
T
b
Mˆ
b
D
'
T
b
‰
T
b
M ‰
b
Œ
'
b
D
'
T
b
Œ
m
b
Œ
'
b
: (3.14)
erefore, the unpacking of Equation (3.14) results in the “body” mode kinetic energy dis-
tribution expressed in terms of generalized coordinates associated with the seven shape patterns
per axial station, i.e.,
Œ
KE
b
D
Œ
m'
b
˝
Œ
'
b
: (3.15)
is result offers the opportunity to describe the “body” modal kinetic energy distributions
in terms of generalized “directions.” Specifically,
a. “body X” bending kinetic energy:
KE
bx;n
D
X
1;8;:::
f
KE
b
g
n
C
X
5;12;:::
f
KE
b
g
n
: (3.16)
b. “body Y ” bending kinetic energy:
KE
by;n
D
X
2;9;:::
f
KE
b
g
n
C
X
4;11;:::
f
KE
b
g
n
: (3.17)
c. “body Z” axial kinetic energy:
KE
bz;n
D
X
3;10;:::
f
KE
b
g
n
: (3.18)
d. “body” torsion kinetic energy:
KE
bt;n
D
X
6;13;:::
f
KE
b
g
n
: (3.19)
e. “body .n D 0/” bulge kinetic energy:
KE
bb;n
D
X
7;14;:::
f
KE
b
g
n
: (3.20)
e sixth “breathing” kinetic energy is described by the previous result in Equation (3.12),
specifically
f. “breathing” kinetic energy:
KE
r;n
D
X
1;2;3;:::
f
KE
r
g
n
: (3.21)
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset