DTFM Modeling and Sensitivity Analysis for Long Masts · DTFM Modeling and Sensitivity Analysis for...
Transcript of DTFM Modeling and Sensitivity Analysis for Long Masts · DTFM Modeling and Sensitivity Analysis for...
May 4, 2004
DTFM Modeling and Sensitivity Analysis for Long Masts
May 4, 2004
Current Status
• Completed formulations for DTFM modeling of long masts
• Initiated MATLAB programming for a multiple-bay mast dynamic analysis
Bending Frequency of Solar Sail Mast (SN002)
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Time (sec)
Tip
Dis
plac
emen
t (in
)
May 4, 2004
Distributed Transfer Function Method
--DTFM decomposes the structure only at those points where multiple structural components are connected minimum number of nodes, small matrices, & high computational efficiency.
--Closed form analytical solutions reliable results.--Able to model local material and geometrical imperfections.--Convenient in handling structural systems with passive and active
damping, gyroscopic effects, embedded smart material layers as sensing and actuating devices, and feedback controllers.
Why DTFM is unique?--In the Laplace domain.--Using Distributed Transfer Function instead of Shape Function.
Why DTFM is distinctively suitable for solar sails?
May 4, 2004
Mast Analysis Using the DTFM
1. Decomposition of a mast into components.2. Generation of state space form for each component.3. Generation of distributed transfer function for each component.4. Generation of dynamic stiffness matrix for each component and
assembly of components.5. Static and dynamic solutions:
• Natural Frequencies and mode shapes.• Buckling analyses.• Frequency Responses.• Static and Dynamic Stress Analyses.• Time Domain Responses.
May 4, 2004
DTFM Mast Analysis: Step 1--DecompositionD
ecompositionA
ssem
bly
May 4, 2004
DTFM Mast Analysis: Step 2--State Space Form
A set of governing equations for each individual component:
a bt
ct
u x tx
f x tijk ijk ijk
kj
kk
N
j
n
i
j
+ +FHG
IKJ =
==ÂÂ ∂
∂∂∂
∂∂
2
201
,,
a f a f
x L t i nŒ ≥ =0 0 1, , , , ,a f L
Example: a beam component
EI vx
A vt
p∂∂
r ∂∂
4
4
2
2+ =
May 4, 2004
DTFM Mast Analysis: Step 2--State Space Form
ddx
x s F s x s q x sh h( , ) ( ) ( , ) ( , )= +State space form:
Example: a beam component
h( , )
( , )( , )( , )( , )
x s
v x sv x sv x sv x s
=¢¢¢¢¢¢
RS||
T||
UV||
W||
F sAs
EI
( ) =-
L
N
MMMMM
O
Q
PPPPP
0 1 0 00 0 1 00 0 0 1
0 0 02r
q x s
p x s EI
( , )
( , )
=
RS||
T||
UV||
W||
000
May 4, 2004
DTFM Mast Analysis: Step 3--DTF
A boundary value problem:ddx
x s F s x s q x sh h( , ) ( ) ( , ) ( , )= + x LŒ( , )0
M s N L s r sh h( , ) ( , ) ( )0 + =
The solution is expressed as transfer functions:
h z z z( , ) ( , , ) ( , ) ( , ) ( )x s G x s q s d H x s r sL
= +z0 x LŒ( , )0
G x s e M Ne Me xe M Ne Ne x
F s x F s L F s
F s x F s L F s L( , , ) ( )
( )
( ) ( )
( ) ( ) ( )( )z z
z
z
z= + £
- + ≥RST
- -
- -
1
1
a f
H x s e M NeF s x F s L( , ) ( )( ) ( )= + -1
May 4, 2004
DTFM Mast Analysis : Step 3--DTF
η α εx s x s x sT T T, , ,a f a f a f=State space vector:
a a a ax s x s x s x sT Tn
T T, , , ,a f a f a f a f= 1 2 L
e e e ex s x s x s x sT Tn
T T, , , ,a f a f a f a f= 1 2 L
Displacement vector:
Strain vector:
s ex s E x s, ,a f a f=Force vector:
Example: a beam component
a( , )( , )( , )
x sv x sv x s
=¢RST
UVWe( , )
( , )( , )
x sv x sv x s
=¢¢¢¢¢RST
UVWs e( , )
( , )( , )
( , )( , )( , )
x sQ x s
M x sE x s
EIEI
v x sv x sf
= RSTUVW = = LNM
OQP
¢¢¢¢¢RST
UVW0
0
May 4, 2004
DTFM Mast Analysis : Step 4--Dynamic Stiffness Matrix
ss
aa
s s
s s
0 0 0 0 00
0
,,
, ,, ,
,,
( , )( , )
sL s
EH s EH sEH L s EH L s
sL s
p sp L s
L
L
a fa f
a f a fa f a f
a fa f
LNMOQP =LNM
OQPLNMOQP +LNMOQP
Force vectors at two ends of the component:
Transformed from distributed external forcesDynamic stiffness matrix
Systematically assembles dynamic stiffness matrices of each component
Dynamic stiffness matrix of the whole system
K U Ps s sa f a f a f× =
May 4, 2004
DTFM Mast Analysis: Step 5--Static and Dynamic Solutions
Resonant frequencies of the structure:
det K sia f = 0 si i= - ¥1 w
Mode shapes--nontrivial solutions:
K Us si ia f a f¥ = 0
Frequency responses:
U K Ps s sa f a f a f= - ¥1
Static analysis:K U P0 0 0a f a f a f¥ =
Time domain responses:
Inverse Laplace transform
May 4, 2004
Examples of DTFM Analyses
(1) Two elastically coupled beams
(2) Sensitivity Analysis of a Light-Weight Gossamer Boom
May 4, 2004
Example (1)--Two elastically Coupled Beams
1 2 3
4 5 6
(1) (2)
(3) (4)
EI=40 ρA=0.5
EI=50 ρA=0.5
k=200 k=400
K( ) *
( )( )( )( )( )( )
~ ( )~ ( )~ ( )~ ( )~ ( )~ ( )
s
v sv sv sv sv sv s
Q sM sQ sQ sQ sM s
f
f
2
2
3
4
5
5
2
2
3
4
5
5
¢
¢
R
S
|||
T
|||
U
V
|||
W
|||
=
R
S
|||
T
|||
U
V
|||
W
|||
May 4, 2004
Example (1)--Two Elastically Coupled Beams
Mode
number
DTFM
6*6 matrix
FEM
18 Elements
FEM
34 Elements
FEM
66 Elements
1 16.3 16.3 16.3 16.3
2 41.0 41.1 41.0 41.0
3 54.6 53.1 54.2 54.5
4 79.2 77.8 78.9 79.1
5 144.7 138.3 143.1 144.3
6 157.0 150.5 155.4 156.6
7 273.9 258.1 269.9 272.9
8 305.2 288.2 289.9 304.1
9 448.7 415.4 440.4 446.6
10 500.5 463.9 491.2 498.1
11 669.1 601.7 653.7 665.3
12 747.5 672.7 730.5 743.3
May 4, 2004
Example ( 2)--Sensitivity Analysis of a Light-Weight Gossamer Boom
Buckling analysis of a boom:2 2 2
2 2 2
d d dEI w(x) P w(x) 0dx dx dx
⎛ ⎞+ =⎜ ⎟
⎝ ⎠EI is not a constant along the boom: Divided the boom into a number of sections and each sections is considered to be uniform—Stepwise uniform
Transfer functions are expressed as :1
1
H(x)M ( ), xG(x, )
H(x)N (L) ( ), x
−
−
⎧ Φ ξ ξ <ξ = ⎨
− Φ Φ ξ ξ >⎩
1H(x) (x)(M N (L))−= Φ + Φ
x x xk k∈ +( , )1Φ Φ( , ) $ ( , ) ( ) ... ( ) ( )( ) ( ) ( ) ( )x s x s e T s e T s e T s eF x xk
F x x F x x F xk k k k k≈ = + −− − −1 1 2 2 1 1 12 1
n nk 1
k 1 k
I 0T C
0 E E×
−+
⎡ ⎤= ∈⎢ ⎥⎣ ⎦
May 4, 2004
Example (2)--Sensitivity Analysis of a Light-Weight Gossamer Boom
Length of the inflatable boom: 197 inches Bending stiffness : 656673 lb in* ^2EI0
0xEI EI (1 sin( ))Lπ
= + ε×
ε 0% ± 2% ± 4% ± 6% ± 8% ± 10%Pcr (+ %) 167.0 169.7 172.7 175.4 178.2 181.1 Pcr (- %) 167.0 164.2 161.2 158.5 155.6 152.8
ε 0% ± 2% ± 4% ± 6% ± 8% ± 10%Pcr/Pcr0 1.0000 1.017 1.034 1.051 1.067 1.085 Pcr/Pcr0 1.0000 0.983 0.966 0.949 0.932 0.915
Buckling force as the function of bending stiffness deviation ε
Ration of buckling force changing as the function of ε
May 4, 2004
DTFM Synthesis for Solar Sails
May 4, 2004
Decomposition of a Solar Sail
Membrane
Spacecraft
Mast
Decomposition
Assembly
May 4, 2004
Dynamic Stiffness Matrix Synthesis
Dynamic stiffness matrices of masts—ready.Dynamic stiffness matrix of the spacecraft—lumped mass, ready.
Steps needed to get dynamic stiffness matrices of membranes:1) PVP membrane analysis2) Laplace transform
Mx Kx f&& + =Ms K x f2 + =d i $ $
Solar sail synthesis:1) Displacement compatibility2) Force balance