Spacecraft StructuresSpace System Design, MAE 342, Princeton University

Robert Stengel

Copyright 2008 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE342.html

• Discrete (lumped-mass)structures

• Distributed structures

• Buckling

• Structural dynamics

• Finite-element analysis

Spacecraft Mounting

for Launch

• Spacecraft protected fromatmospheric heating andloads by fairing

• Fairing jettisoned whenatmospheric effects becomenegligible

• Spacecraft attached torocket by adapter, whichtransfers loads between thetwo

• Spacecraft (usually)separated from rocket atcompletion of thrusting

• Clamps and springs forattachment and separation

Fairing Constraints for

Various Launch Vehicles

• Static envelope

• Dynamic envelope accountsfor launch vibrations, withsufficient margin for error

• Various appendages stowedfor launch

• Large variation inspacecraft inertialproperties whenappendages are deployed

STEREO Spacecraft Primary

Structure Configuration

• Spacecraft structuretypically consists of

– Beams

– Flat and cylindrical panels

– Cylinders and boxes

• Primary structure is the“rigid” skeleton of thespacecraft

• Secondary structure maybridge the primary structureto hold components

Solar TErrestrial RElations Observatory

UARS Primary and

Secondary Structure

• Instrument Module provides

– Support for 10 scientificinstruments

– Maintains instrumentalignment boresights

– Interfaces to launch vehicle(SSV)

• Secondary Structuresupports

– 6 equipment benches

– 1 optical bench

– Instrument mounting links

– Solar array truss

– Several instruments havekinematic mounts

Structural Material Properties

• Stress, !: Force per unit area

• Strain, ": Elongation per unit length

!

" = E #

• Proportionality factor, E: Modulus of elasticity, or Young!s modulus

• Strain deformation is reversible below the elastic limit

• Elastic limit = yield strength

• Proportional limit ill-defined for many materials

• Ultimate stress: Material breaks

• Poisson!s ratio, #:

!

" =#lateral

#axial,

typically 0.1 to 0.35

• Thickening under compression

• Thinning under tension

Uniform Stress Conditions

• Average axial stress, !

!

" = P A = Load Cross Sectional Area

• Average axial strain, "

!

" = #L L

• Effective spring constant, ks

!

" = P A = E# = E$L

L

P =AE

L$L = k

s$L

Springs and Dampers

!

fx = "ks#x = "ks x " xo( ) ; k = springconstant

!

fx = "kd#˙ x = "kd#v = "kd v " vo( ) ; k = dampingconstant

• Force due to linear spring

• Force due to linear damper

Mass, Spring, and Damper

!

"˙ ̇ x = fx m = #kd"˙ x # ks"x + forcing function( ) m

!

"˙ ̇ x +kd

m"˙ x +

ks

m"x =

forcing function

m

!

"˙ ̇ x + 2#$n"˙ x +$

n

2

"x =$n

2

"u

!

"n = natural frequency, rad /s

# = damping ratio

$x = displacement

$u = disturbance or control

• Newton!s second law leads to asecond-order dynamic system

Response to Initial Condition

• Lightly dampedsystem has adecaying, oscillatorytransient response

• Forcing by step orimpulse produces asimilar transientresponse

!

"n

= 6.28 rad /sec

# = 0.05

Oscillations

!

"x = Asin #t( )

!

"˙ x = A# cos #t( )

= A# sin #t + $ 2( )

!

"˙ ̇ x = #A$ 2sin $t( )

= A$ 2sin $t + %( )

• Phase angle of velocity (wrt displacement) is $/2 rad (or 90°)

• Phase angle of acceleration is $ rad (or 180°)

• As oscillation frequency, %, varies

– Velocity amplitude is proportional to %

– Acceleration amplitude is proportional to %2

Response to

Oscillatory Input

!

A" 2sin "t + #( )[ ] + 2$"

nA" sin "t + # 2( )[ ] +"

n

2

Asin "t( )[ ] ="n

2

Bsin "t( )[ ]

• ... however, A must be a complex number for this to work

• A better way: Compute Laplace transform to find transferfunctions

!

L "x(t)[ ] = "x(s) = "x(t)e#stdt0

$

% , s =& + j', ( j = i = #1)

!

L "˙ x (t)[ ] = s"x(s)

L "˙ ̇ x (t)[ ] = s2"x(s)

• Neglecting initial conditions

Transfer Function

• or

!

L "˙ ̇ x + 2#$n"˙ x +$

n

2

"x( ) = L $n

2

"u( )

!

s2 + 2"#

ns+#

n

2( )$x(s) =#n

2$u(s)

• Transfer function from input to displacement

!

"x(s)

"u(s)=

#n

2

s2 + 2$#

ns+#

n

2( )

Transfer Functions of Displacement,

Velocity, and Acceleration

!

"x(s)

"u(s)=

#n

2

s2 + 2$#

ns +#

n

2( )"˙ x (s)

"u(s)=

#n

2s

s2 + 2$#

ns +#

n

2( )"˙ ̇ x (s)

"u(s)=

#n

2s

2

s2 + 2$#

ns +#

n

2( )

• Input tovelocity:multiply by s

• Input toacceleration:multiply by s2

• Transfer functionfrom input todisplacement

From Transfer Function to

Frequency Response

• Displacement frequency response (s = j%)

!

"x( j#)

"u( j#)=

#n

2

j#( )2

+ 2$#n j#( ) +#n

2!

"x(s)

"u(s)=

#n

2

s2 + 2$#

ns+#

n

2( )

• Displacement transfer function

Real and imaginary components

Frequency Response

• %n: natural frequency of the system

• %: frequency of a sinusoidal input to the system

!

"x( j#)

"u( j#)=

#n

2

j#( )2

+ 2$#n j#( ) +#n

2

=#n

2

#n

2 %# 2( ) + 2$#n j#( )&

#n

2

c #( ) + jd #( )

=#n

2

c #( ) + jd #( )

'

( )

*

+ , c #( ) % jd #( )c #( ) % jd #( )

'

( )

*

+ , =

#n

2c #( ) % jd #( )[ ]

c2 #( ) + d2 #( )

& a(#) + jb(#) & A(#)e j- (# )

• Frequency response is a complex function

– Real and imaginary components, or

– Amplitude and phase angle

Frequency

Response of the

2nd-Order System

• Bode plot– 20 log(Amplitude Ratio) [dB] vs. log !

– Phase angle (deg) vs. log !

• Natural frequency characterized by

– Peak (resonance) in amplituderesponse

– Sharp drop in phase angle

• Acceleration frequency responsehas the same peak

• Convenient to plot response onlogarithmic scale

!

ln A(")e j# (" )[ ] = lnA(") + j#(")

Acceleration

Response of the

2nd-Order System

• Important points:– Low-frequency acceleration

response is attenuated

– Sinusoidal inputs at naturalfrequency resonate, I.e.,they are amplified

– Component naturalfrequencies should be highenough to minimizelikelihood of resonantresponse

Spacecraft Stiffness* Requirements

for Primary Structure

* Natural frequency

Example 8.1(Fundamentals of Space Systems, 2005)

• Atlas IIAS launch vehicle

• Spacecraft structure meetsprimary stiffnessrequirements

• What are axial stiffnessrequirements for Units Aand B?

– Support deck naturalfrequency = 50 Hz

Octave Rule: Componentnatural frequency ! 2 xnatural frequency ofsupporting structure

• Unit A: 2 x 15 Hz = 30 Hz, supported by primary structure

• Unit B: 2 x 50 Hz = 100 Hz, supported by secondary structure

Factors and Margins of Safety

• Factor of Safety– Typical values: 1.25 to 1.4

!

Allowable load (yield stress)

Expected limit load (stress) "Design factor of safety#1

• Margin of Safety– “the amount of margin that exists above the

material allowables for the applied loadingcondition (with the factor of safety included)”,Skullney, Ch. 8, Pisacane, 2005

!

Load (stress) that causes yield or failure

Expected service load

Worst-Case Axial Stress

on a Simple Beam

• Axial stress due to bending

!

" = My I

!

" =M h /2( )

I

• Maximum stress

• Worst-case axial stress due to bending and axial force

!

"wc

= ±P

A

#

$ %

&

' ( max

±M h /2( )

I

Example 8.3(Fundamentals of Space Systems, 2005)

• Spacecraft weight = 500 lb

• Atlas IIAS launch vehicle

• Factor of safety = 1.25

• Maximum stress onspacecraft adapter?

Atlas IIAS Limit Loads (g)

Example 8.3, con’t.

• Worst-case axial load at BECO (5±0.5 g)

!

"wc

= ±P

A

#

$ %

&

' ( max

±Mc

I

!

A = 2"rt = 7.1 in2

I = "r3t = 286 in4

• Worst-case stress

!

"wc =500 # 5.5

7.1+500 # 0.5 # 42 # 9

286

$

% & '

( ) #1.25 = 897.1 psi

• Worst-case lateral load at BECO (2.5 ± 1 g) or Maximum FlightWinds (2.7 ± 0.8 g)

!

"wc =500 # 3.5

7.1+500 # 2 # 42 # 9

286

$

% & '

( ) #1.25 =1960 psi

Force and Moments on a Slender

Cantilever (Fixed-Free) Beam

• Idealization of

– Launch vehicle tied-down to alaunch pad

– Structural member of apayload

• For a point force

– Force and moment must beopposed at the base

– Shear distribution is constant

– Bending moment increases asmoment arm increases

– Torsional moment andmoment arm are fixed

Structural Stiffness

• Geometric stiffening property of a structure is portrayed by thearea moment of inertia

• For bending about a y axis (producing distortion along an x axis)

• Area moment of inertia for simple cross-sectional shapes

• Solid rectangle of height, h, andwidth, w:

• Solid circle of radius, r:

• Circular cylindrical tube with innerradius, ri: and outer radius, ro:

!

Iy = wh3/12

!

Iy = "r4 /4

!

Iy = " ro4

# ri4( ) /4

!

Ix

= x z( )z2 dzzmin

zmax

"

Bending Stiffness

• Neutral axis neither shrinks nor stretches in bending

• For small deflections, the bending radius of curvature ofthe neutral axis is

!

r =EI

M

• Deflection at a point characterized bydisplacement and angle:

Bending

Deflection

• Second derivative of z and first derivative of & areinversely proportional to the bending radius:

!

d2z

dx2

=d"

dx=My

EIy

Maximum Deflection and

Bending Moment of Beams(see Fundamentals of Space Systems for additional cases)

Fixed-Free Beam Fixed-Fixed Beam Pinned-Pinned Beam

Ymax = maximum deflection Mmax = maximum bending moment

Maximum Deflection and

Bending Moment of Plates(see Fundamentals of Space Systems for additional cases)

Circular Plate

Rectangular Plate!

m =1/"

Typical Cross-Sectional Shear Stress

Distribution for a Uniform Beam

• Shear stress due to bending moment ishighest at the neutral axis

• Maximum values for various cross sectons(see Fundamentals of Space Systems)

Buckling

• Predominant steady stressduring launch is compression

• Thin columns, plates, andshells are subject to elasticinstability in compression

• Buckling can occur below thematerial!s elastic limit

!

"cr

=C# 2

E

L /$( )2

=P

A

Critical buckling stress of a column(Euler equation)

!

C = function of end " fixity"

E = modulus of elasticity

L = column length

" = I A = radius of gyration

Pcr = critical buckling load

A = cross sectional area

Effect of “Fixity” on Critical

Loads for Beam Buckling

• Euler equation

– Slender columns

– Critical stress below theelastic limit

– Relatively thick column walls

• Local collapse due to thinwalls is called crippling

Crippling vs. Buckling

Critical Stress for Plate and Cylinder Buckling

Bending Moment and Linear Deflection

due to a Distributed Normal Force

• Deflection is found by four integrations of the deflectionequation

!

d2

dx2EIy

d2z

dx2

"

# $

%

& ' x= xs

= N 'y xs( )

N!y(xs)

!

My x( ) = Ny (x)xmin

xmax

" x # xcm( )dx

= N 'y (x)xmin

xmax

" dx x # xcm( )dxxmin

xmax

"

!

N '(x) = normal force variation with length

Bending Vibrations of a

Free-Free Uniform Beam

!

EIyd4z

dx4

x= xs

= k = "m'd2z

dt2

x= xs

!

EIy = constant

m'= mass variation with length (constant)

k = effective spring constant

• Solution by separation of variables requires that left andright sides equal a constant, k

• An infinite number of separation constants, ki, exist

• Therefore, there are an infinite number of vibrationalresponse modes

Bending Vibrations of a

Free-Free Uniform Beam

• In figure, (u = z, y = x)

• Left side determinesvibrational mode shape

• Right side describesoscillation

• Natural frequency ofeach mode proportionalto (ki)

1/2

Mode shapes of bending vibrations

!

EIyd4z

dx4

= ki = "m'd2z

dt2

Fundamental Vibrational

Frequencies of Circular Plates

f = natural frequencyof first mode, Hz

Finite-Element

Structural Model

TIMED Spacecraft • Grid of elements, each with

– mass, damping, andelastic properties

– 6 degrees of freedom

Vibrational Mode Shapes

for the X-30 Vehicle (Raney

et al, J. Aircraft, 1995)

Computational Grid Model

• Body elastic deflection distorts the shape ofscramjet inlet and exhaust ramps

• Aeroelastic-propulsive interactions

• Impact on flight dynamics

Shapes of the First Seven Modes

Next Time:Flight Path Guidance,

Navigation, and Control