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