Thermal Fatigue Analysis

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O R IG IN A L PA PE R

Thermal fatigue analysis of small-satellite structure

Gasser Farouk Abdelal Nader Abuelfoutouh Ahmed Hamdy Ayman Atef

Received: 11 November 2006 / Accepted: 12 January 2007 / Published online: 16 February 2007 Springer Science+Business Media B.V. 2007

Abstract The small-satellite thermal subsystemmain function is to control temperature ranges on

equipments, and payload for the orbit specified.

Structure subsystem has to ensure the satellitestructure integrity. Structure integrity should

meet two constraints; first constraint is accepted

fatigue damage due to cyclic temperature, andsecond one is tolerable mounting accuracy at

payload and Attitude Determination and Control

Subsystem (ADCS) equipments seats. First,

thermal analysis is executed by applying finite-difference method (IDEAS) and temperature

profile for satellite components case is evaluated.Then, thermal fatigue analysis is performed

applying finite-element analysis (ANSYS) tocalculate the resultant damage due to on-orbit

cyclic stresses, and structure deformations at the

payload and ADCS equipments seats.

Keywords Aerospace Satellite Structure Finite difference Finite element Thermal Mounting accuracy Fatigue

1 Introduction

Structure made of different materials can experi-

ence thermal stresses without external constraintseven under uniform temperature. Differences in

the materials coefficients of thermal expansionproduce incompatible strains and resulting ther-mal stresses. These stresses balance when no

external constraints are present.Thermally induced loads and stresses are

limited by deformation; once a material reaches

its proportional limit, or once the structure beginsto buckle, thermal stress no longer increases in

proportion to the change in temperature. Ductile

materials seldom rupture or buckle from a singleapplication of thermal stress, but they can fail in

fatigue from the many cycles of thermal loadingcommon to orbiting satellite.

Actual thermal-design problems for satellite

are complicated. The design problem typically

must combine multiple modes of heat transferwith time varying boundary conditions that

require transient instead of steady-state solutions.

To predict satellites temperatures, the thermalanalysis problem combines two types of math

models. The first one is a thermal-radiation model

G. F. Abdelal (&) A. Hamdy A. AtefEgyptian Space Program, National Authority forRemote Sensing and Space Science (NARSS), 23Jozef Broz Teto Street, Elnozha Elgededah, Cairo11769, Egypte-mail: [email protected]

N. AbuelfoutouhAerial Photograph Division Program, NationalAuthority for Remote Sensing and Space Science(NARSS), 23 Jozef Broz Teto Street, ElnozhaElgededah, Cairo 11769, Egypt

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Int J Mech Mater Des (2006) 3:145159

DOI 10.1007/s10999-007-9019-1


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to calculate external heating rates by simulatingthe external geometry of the satellite including

surface properties. By subjecting this model to a

simulated orbit, the output from this modeconsisting of; the environmental heating rates

due to direct solar, albedo, and planetary emis-

sions; and external radiation between satellitesurfaces; becomes input for the second model.

The second thermal model uses a thermal

analyzer. The satellite is modeled much the sameas the structural analysis model with internal

details but the analysis is based on the finite

difference method. Then the heating sources aredefined. Finally, the model is solved to simulate

the heat transfer paths of conduction, convec-

tion, and radiation. The thermal analyzer calcu-lates temperatures at all nodes for steady-state

or transient conditions by solving energyequations.

A thermoelastic analysis for Small Sat struc-

ture due to on-orbit cyclic temperature is consid-

ered. It calculates stresses and distortions due totemperatures cyclic loading. The results of ther-

moelastic analysis are used to calculate fatigue

damage due to on-orbit cyclic stresses. Moreover,it is used to check mounting accuracy of the

precise equipments (MBIE and ADCS equip-

ments) after on-orbit thermal deformation. The

most critical structural module in Small Satregarding by on-orbit thermal deformation is the

basis unit block module. This module is verysensitive to thermal deformation because it

carries the precise equipments of the satellite.

The rest of structural modules do not havesevere restrictions on their equipment mounting

accuracy.

2 Satellite overview

The satellite configuration is shown below inFig. 1. The structure is a 1.03 0.8 1.0255 m

cube with four solar panels attached to the

satellite by means of rotation mechanism. Fourheat shields are installed on the satellite structure

to prevent internal instruments from direct envi-

ronmental heat loads. The highest power con-suming components will be placed away from hot

heat shields subjected to solar radiation. The

primary main elements of the satellite structureare shown in Fig. 2.

3 Mission requirements and constraints

The temperature constraint is specified for each

component. Table 1 lists temperature ranges for

different satellite components. Component tem-perature constraints contain an upper and a

4 Solar panels

4 Heat shields

Satellite case

Fig. 1 The satellite under study

Fig. 2 3-D model of satellite structure

Table 1 Satellite component temperature limits

Component Temperature constraints

Power subsystem 5 to 45CCommunication subsystem 5 to 45CAttitude control subsystem 5 to 45COnboard computer 0 to 40CSolar panel 100 to 85CStructure 100 to 100C

146 Int J Mech Mater Des (2006) 3:145159

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lower bound of each components preferredoperating temperature range. The thermal

design of the satellite must not allow any

component to exceed absolute operatingtemperature ranges. Components operate most

efficiently and effectively when preferred oper-

ating temperatures are maintained. In addition,component life span will be degraded by oper-

ating the component outside of its preferred

temperature range.The satellite orbits Earth at the following

orbital parameters:

Sun-synchronous orbit

Altitude: 576 km

Inclination angle: 97.721

Local time of descending node: 9 h: 45 min

b Angle is defined as the angle between the

solar vector and the normal to the orbital plane.According to the orbital parameters listed above,

Beta angle (b) can be calculated mathematicallyas (Gilmore 1994):

b sin1cos dS sin i sinXXS sin dS cos i1

Where, dS = declination of the sun; i = orbit

inclination; (W WS) = local time of descendingnode.

Figure 3 shows the b angle history during one

year. b Angle will reach its maximum value

(about 63) at summer solstice while the min-imum value will be at winter solstice (about

55).

4 Thermal model

Thermal analysis involves constructing a geomet-ric math model (GMM) and a thermal math

model (TMM) of the satellite and identifying

analysis cases to be run. The GMM and the TMM

serve different purposes. The GMM is a mathe-matical representation of the physical surfaces of

the satellite and is used to calculate black bodyradiation couplings between surfaces as well as

heating rates due to environmental fluxes. TheTMM is most often a lumped parameter network

representation of the thermal mass and conduc-

tion and radiation couplings of the satellite, and isused to predict satellite temperatures. The radi-

ation interchange couplings and environmental

heat fluxes calculated by the GMM are used in

constructing the TMM. Both the GMM andTMM are constructed and executed using I-

DEAS TMG. First, the GMM of the satellitewas constructed using I-DEAS TMG. The model

consists of a simple representation of the satellite

and the payload. It was constructed using rectan-gular, circular, and cylindrical surfaces, and each

surface was assigned the appropriate solar

absorptivity and emissive. Dimensions and massof each component of the satellite can be found in

Table 2.

A transient-state analysis was conducted to getthe temperature variation of each component

with during one complete orbit.

The effects of the Beta angle are considered byusing different possible orbital parameters. This

angle plays an important roll in determining the

eclipse and sunlight time periods for the satellite

Fig. 3 b angle historyduring 1 year

Int J Mech Mater Des (2006) 3:145159 147

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and hence irradiation due to different sources

such as the Sun, Earth and albedo.

Black paint is used to coat the electronic boxesand on the inside surfaces to improve energy

exchange with the other subsystems onboard the

satellite. Moreover, the external structure iscoated by white paint to minimize solar radiation

absorption. Finally, Multilayer insulation (MLI)

is used to control the temperature of the payload.A catalog of the thermal properties of each

material used on the satellites is listed in Table 3.

The element chosen from the I-DEAS TMGlibrary is linear quadrilateral thin shell element.

Table 2 Number of nodes and elements for some selectedstructural modules

Satellite structuremodule

No. of elements No. of nodes

Basis plate 146 242Basis walls 66 104

Star sensor bracket 32 61

Table 3 Mass, material properties, and thermo-optical properties of the different coatings applied

Subsystem and Component Mass (kg) Material Coating

PayloadMulti-band earth imager 45 Titanium alloys Chemglaze Black paint Z306 (inside)

Multi-layer Insulation (outside)

Payload CDHS unit 7.2 Aluminum 2024-Tx Chemglaze White paint A276MEI signal processing unit 3.7 each Aluminum 2024-Tx Chemglaze White paint A276

ADCSStar sensor 4 Aluminum 2024-Tx Chemglaze Black paint Z306Angular velocity meter Gyro 1 each Aluminum 2024-Tx Chemglaze White paint A276Interface unit for each gyro 0.92 each Aluminum 2024-Tx Chemglaze White paint A276Magnetometer 1.5 Aluminum 2024-Tx Chemglaze White paint A276Magnetorquer 0.38 each Aluminum 2024-Tx Chemglaze White paint A276Reaction wheel 3.3 each Aluminum 2024-Tx Chemglaze White paint A276

Communications subsystemX-band equipment

X-band electronic module 3.8 Aluminum 2024-Tx Chemglaze White paint A276X-band antenna 1.6 Aluminum 2024-Tx Uncoated

S-band equipmentS-band electronic module 2.2 each Aluminum 2024-Tx Chemglaze White paint A276S-band conical antenna 0.27 each Aluminum 2024-Tx UncoatedS-band dipole antenna 0.13 Aluminum 2024-Tx Uncoated

GPS receiverGPS electronic module 1.1 Aluminum 2024-Tx Chemglaze White paint A276GPS antenna 0.15 each Aluminum 2024-Tx Uncoated

Platform CDHSOn-board digital computing complex 3.7 each Aluminum 2024-Tx Chemglaze White paint A276Telemetry module 2.8 Aluminum 2024-Tx Chemglaze White paint A276

Power SubsystemBattery cell module 16.5 Ni-Cd Chemglaze Black paint Z306Power-conditioning unit (PCU) 3.6 Aluminum 2024-Tx Chemglaze White paint A276Cells leveling unit (CLU) 1.9 Aluminum 2024-Tx Chemglaze White paint A276

Solar array panels 6.8 Gallium arsenide Cell side: as = 0.75, e = 0.9 Backside:Chemglaze Black paint Z306

Thermal subsystemHeat shields 3.6 Honeycomb structure Chemglaze Black paint Z306Multi-layer Insulation 0.2

Structure and mechanism subsystemSatellite structural modulesRotation mechanism 41 Aluminum 2024-Tx Chemglaze Black paint Z306Locking and releasing mechanism 1.7 Aluminum 2024-TxSeparation transducer 0.5 Aluminum 2024-Tx

0.1 Aluminum 2024-TxSatellite total mass 205


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This type of elements assumes a constant thick-

ness. Because of the small thickness of all theelements, this thin shell was the most appropriate

element which could represent it without using an

excessive number of elements. The thermalmodel consists of 8,574 shell elements with

different thicknesses and 1,0138 nodes. Figure 4

shows the structure of the satellite after completemeshing and Table 2 define the mesh character-

istics for some selected elements.

4.1 Modeling parameters

The I-DEAS TMG software uses a finite differ-

ence method to solve the heat balance equation

and to get the temperature distribution in themodel. The heat balance equation for a transient

run to be solved iteratively at iteration n + 1 and

time t + Dt for element i can be cast in the form(Camack and Edwards 1960):

ci;t Ti;tDt;n1Ti;t Dt

Qi;tDttasAsqsi;tDttaAAqAi;tDtteAEqEi;tDtt

rXNj1

FijAij T4j;tDt;n1T4i;tDt;n1

XN

j

1

Kij Tj;tDt;n1Ti;tDT;n1 2where, Ci,t = the capacitance of the element;Ti, t + dt, n + 1 = the temperature of element i at

time t + dt and iteration n + 1; n = the current

iteration value; Ti,t = the temperature of elementi at time t; Dt = the integration time step;

Qi,t + dt(t) = the heat generation rate of element

i as a function of time at time t + dt; Gij = the sumof the conductive conductances between i and j; r

= Stefan Boltzmann constant; Fij = gray body

view factor between elements i and j; A =radiating surface area of element i.

The Backward method was used to solve Eq.

(2). In particular, it is most effective for thismodel where s min is small compared to the

solution interval. In addition, it is more accuratethan the ForwardBackward method under

conditions of rapid temperature change.

Listing of Transient Analysis Parameters,

1. Start Time = 0.0 s.2. End Time = 116686.8 s.

3. Integration Method: Backward.4. Time Step value = 58.8 s.5. Maximum Number of Iterations = 500.

6. Relaxation Factor = 0.05.

7. Temperature Difference for Convergence =0.1.

Listing of Solution Methods,

1. Solution Method: Conjugate Gradient2. Iteration Limit = 300.

Fig. 4 Satellite structure after meshing on I-DEAS TMG

Fig. 5 Shown temperature distribution of base plate


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3. Convergence Criterion = 0.1.

4. Preconditioning Matrix Fill value = 10.

5 Thermal analysis results

Satellite heating and cooling occurs while it goes

through sunlight and eclipse respectively. Maxi-mum and minimum temperatures for the satellite

will be calculated for the orbits sunlight and

eclipses zones. Temperature variations of basisplate and basis walls during one orbit are shown

in Figs. 6 and 7. Table 4 shows maximum andminimum temperatures for various satellite com-

ponents.

6 Thermo-elastic analysis

The most critical structural module in Small Sat

affect with on-orbit thermal deformation is the

basis unit block module. This module is verysensitive for the thermal deformation because it

carries the precise equipments into the satellite,

Fig. 6 Temperaturevariations of basis wallsduring one orbit

Fig. 7 Temperaturevariations of basis plateduring one orbit


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otherwise, the rest of structural modules have notbig restrictions on their equipments mounting

accuracy. The following points are taken into

consideration during building this model:

I. Basis unit block module consists of basisplate, four basis walls, two diagonal struts,

and star sensor bracket.

II. The material is Aluminum alloy AMg6(E = 7.2 103 MPa, m = 0.33, q = 2,630

kg/m3, ry = 150 MPa, ru = 310 MPa). For

thermal analysis, it has the followingproperties:

III. Thermal conductivity (K) = 117 W/m.C

IV. Coefficient of thermal expansion (a) =24.7 106 m/m/C

V. Assign the location of mounting the high

precise equipments located at basis unit

block module to calculate thermal defor-

mation at their mounting points and hence

their mounting accuracy.VI. Equipment support is designed to elimi-

nate thermal loading as they allow the

sliding of this support under thermal

expansion. Therefore, in the simplifiedthermal model all satellite equipment is

removed to avoid the appearance of anyartificial thermal stresses that may result

from the improper representation of the

sliding support.VII. Basis unit block module is meshed with

35,015 elements for the basis plate and49,096 elements for the basis walls, while

the rest of structural modules are meshed

with coarse meshing. SOLID92 elements

are used during meshing process.VIII. The nodes located at connection areas

between the different structural modulesare coupled at all degrees of freedom.

Figure 8 shows the entire finite element model

of Small Sat used during on-orbit thermal defor-

mation analysis.Thermal deformations are evaluated for the

worst cases, which are defined by the on-orbitthermal analysis for the satellite structure. Ther-

mal satellite engineer performs a complete on-orbit thermal analysis of the satellite and supplyinput data for surface temperatures gradient of

different satellite structure modules. These data

are considered as an input data to perform on-orbit thermal deformation analysis. For Small Sat,

the satellite structure is solved thermally due to

on-orbit Input data for surface temperaturesgradient of different satellite structure modules

of Small Sat are listed in Table 2. It contains

maximum and minimum average temperatures

for each structural module.Displacement boundary conditions are usually

specified at model boundaries to define rigidsupport points. During on-orbit operation, the

satellite is totally free without any fixation points,

but to conduct a thermo-elastic analysis, thedisplacement boundary condition must be defined

for the related model. Therefore, to simulate the

satellite thermal deformation due to on-orbitcycling, the satellite model must be constrained

Table 4 Surface temperatures of different satellite struc-ture modules

Satellitestructuremodule

Corneror Face

Maximumtemperature,( C)

Minimumtemperature,( C)

Base plate I 5.3 10.1

II 1.8 6.2III 7.2 4.9IV 5.4 1.6

Mounting plate I 22 11.9II 4.5 7.3III 21.2 9.4IV 18.6 7.5

Basis plate I 17 9.5II 19.8 4.7III 17 10.6IV 14.6 12

Basis walls I-II 32.8 31.3II-III 21.9 20.2

III-IV 21.9 20.2IV-I 21.9 20.2Upper frame I-II 36.1 35

II-III 28.5 29.6III-IV 32.5 31.7IV-I 36.3 35.2

Lower frame I-II 18.5 16.3II-III 12 5.9III-IV 17.6 12.8IV-I 13.5 7.2II 14.7 10.3III 15.2 8

Diagonal strut I 14.3 13.2IV 16.6 15.5

Star sensor bracket 18.8 17.4


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by rigid support points. Selection of these pointslocations and their fixation manner should

provide minimum effect on satellite deformation.

For Small Sat model represented at Fig. 8, thesupport points are selected at plan of star sensor

mounting with its bracket. This plan is selected

because the mounting accuracy of all preciseequipments installed at the basis unit block is

measured relative to the star sensor bracket.

Displacement boundary conditions are applied byfixing one of the star sensor four points of fixation

(on bracket) in X direction. Then, fix the next

point in the Y and one of the other in the Z.By reviewing the input data listed at Table 2,

some of the satellite structural modules aredivided into more than one division according to

their position. Each division has its maximum and

minimum average surface temperature. Beforeconduct a thermo-elastic analysis, a thermal anal-

ysis process is performed to expand temperatures

through all solid elements and calculate temper-ature distribution for all nodes. This process is

performed twice for the worst cases, maximum

and minimum average surface temperatures.A thermo-elastic analysis is conducted to the

entire satellite model based on the output results

from the thermal analysis. The analysis is per-

formed twice for the worst cases, maximum andminimum average surface temperatures. Dis-

placement boundary conditions are applied as

shown in Fig. 8.Thermal strains are given by a. (T-Tref),

where a is the coefficient of thermal expansion, T

is the current element temperature, and Tref is thereference (ambient) temperature. In thermo-elas-

tic analysis, the reference temperature is the zero

thermal stresses temperature which is the assem-bly room temperature (=20C) in Small Sat case.

7 Thermo-elastic analysis results

Temperature distribution for the entire satellite isdetermined as a result of the thermal analysis.

Fig. 8 F.E model used during on-orbit thermal deforma-tion analysis

Fig. 9 Temperature distribution for the entire satelliteduring maximum average surface temperatures

Fig. 10 Temperature distribution for the entire satelliteduring minimum average surface temperatures


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Figures 9 and 10 show the temperature distribu-

tion (C) for the entire satellite during maximumand minimum average surface temperatures.

Stress-strain state of the basis unit block

structure module is determined as a result of theon-orbit thermal deformation (thermo-elastic)

analysis for both worst cases, maximum and

minimum average surface temperatures. Thediagrams of equivalent stress distribution

(103 MPa) and displacements (mm) of this struc-

tural module are shown in Figs. 1114. The stress

values are determined according to Von-Missescriterion. Displacements are relative to the points

of star sensor attachment to its bracket, which arethe fixation points for the entire model of Small

Sat structure during thermo-elastic analysis as

presented at Fig. 6.The maximum stresses values re for basis unit

block module at each on-orbit thermally worst

load cases and there equivalent yield margin ofsafety (MSy) are given in Table 5.

8 Mounting accuracy due to on-orbit thermaldeformation

From static point of view, the basis unit block

module is safe because the yield margin of safetyis a positive value in both of worst design cases.

But this is not enough to determine that the

design is applicable and can withstand to on-orbitthermal loads. Satisfactory performance of the

satellite requires accurate predictions of thermal

Fig. 11 Basis unit block stress distribution due to on-orbitthermal deformation at maximum average surface tem-peratures

Fig. 14 Basis unit block displacement due to on-orbitthermal deformation at minimum average surface temper-atures

Fig. 12 Basis unit block displacement due to on-orbitthermal deformation at maximum average surface tem-peratures

Fig. 13 Basis unit block stress distribution due to on-orbitthermal deformation at minimum average surface temper-atures


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deformations to verify pointing and alignmentrequirements for Mounting Accuracy. Therefore,

it is important to calculate the angular positioning

deviations for all high precise equipments (pay-load and ADCS devices) relative to the star

sensor due to on-orbit thermal deformation.

These values must not exceed the relative limitingdeviations specified for mounting the precise

equipments. Table 6 lists the limiting angular

positioning deviations for the most precise equip-ments relative to the star sensor which are

derived from the structure requirements.

The angular positioning deviation between anytwo equipments is calculated by measuring the

deviation angle between normal vectors for their

mounting plans. The deviation angle is deter-mined by subtracting the measured angle after

on-orbit thermal deformation from the initialangle before deformation. The criterion used tocalculate the angular positioning deviation

between the precise equipments and the star

sensor is explained below. Figure 15 showsgraphical sketch to explain the criterion used to

calculate the angular positioning deviation be-

tween any two equipments.

All precise equipments mounted on basisunit block are fixed through three or four

fixation points. The mounting plan can be

defined by only two vectors connecting at leastthree fixation points. In case of equipment A,

V*

1A, and V*

2A identify its mounting plan before

on-orbit thermal deformation. While, V*

1B andV*

2B identify the mounting plan of equipment B.

The normal vectors, V*

and V*

nB for mounting

plan of equipment A and B respectively beforeon-orbit thermal deformation, can be calculated

by applying vector cross product as following:

V*

nA V*

1A V*

2A

V*

nB

V*

1B

V*

2B

The angle between both of these normal vectorsis calculated by the next formula:

h cos1

V*

1A V*

2A

j V*

1Aj j V*

2Aj

!

After on-orbit thermal deformation, the equip-

ment mounting plan is usually deformed. And so

the normal vectors, V*

nA and V*

nB, are modified toV* 0nA and V* 0nB respectively. They are calculatedfor the deformed mounting plan as following:

V* 0

nA V* 0

1A V* 0

2A

V* 0

nB V* 0

1B V* 0

2B

Where: V* 0

1A & V* 0

2A and V* 0

1B & V* 0

2B identifythe mounting plan of equipment A and B

respectively after on-orbit thermal deformation.

These vectors are calculated from the displace-ment deformation results of on-orbit thermal

deformation analysis in X, Y, and Z direction.

The modified angle between normal vectors for

the deformed mounting plan is calculated asfollowing:

h0 cos1 V* 0

1A V* 0

2A

j V* 0

1Aj j V* 0

2Aj

0@

1A

Table 5 Maximum Von-Misses equivalent stresses andyield margins of safety for basis unit block module at eachon-orbit thermally worst load cases

Design case re (MPa) MSy

Case 1 94.158 0.59Case 2 125.937 0.19s Case 1 represents the thermal deformation due to

on-orbit maximum temperatures.s Case 2 represents the thermal deformation due to

on-orbit minimum temperatures.

Table 6 Limiting angular positioning deviations for themost precise equipments relative to the star sensor

Equipment Limiting angular positioningdeviations, (arcmin)

MBEI (optical-mechanical unit)

30

Angular velocitymeters (gyro)

60

Reaction wheels 60Magnetometer 60Magnetorquers 60


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The angular positioning deviation angle between

equipment A and B is calculated by the next

formula:

hdev h0 h

For Small Sat, the angular positioning devia-

tion angle hdev (arc-min) is calculated for theprecise equipments mounted on basis unit block

relative to the star sensor. Table 7 lists the valuesof angular positioning deviation angle due to on-

orbit thermal deformation relative to the star

sensor. It shows the results for both maximum

and minimum on-orbit temperatures. By compar-ing these results with the limiting values listed at

Table 6, the performance of the satellite is notaffected with on-orbit thermal deformation under

maximum or minimum temperatures.

9 Fatigue damage due to on-orbit thermal cycling

One of the most essential aspects during perform-

ing on-orbit thermal deformation analysis is to

evaluate the fatigue damage due to on-orbit cyclicthermal stresses. Ductile material (AMg6 alumi-

num alloy), used to manufacture the satellite

structure modules, does not rupture or bucklefrom a single application of thermal stress, which is

clear from the analysis until now. But the material

can fail in fatigue from the many cycles of on-orbitthermal deformation loading. The results data of

thermo-elastic analysis is used to evaluate the

thermal fatigue damage for the basis unit block.

By reviewing the results data of thermal

deformation analysis and the thermal stresses

distribution for the basis unit block module undermaximum and minimum temperatures (Figs. 11

14), the maximum thermal stress is occurred atthe same locations under both of the worst cases.

These locations are illustrated in Fig. 16. The

entire satellite structure is affected by a cyclicthermal stresses along the operation life time of

the satellite. However, these points are the most

locations that are affected by fatigue damage dueto on-orbit thermal cycling.

Thermal fatigue damage must be calculated for

the critical points located at basis plate moduleand basis walls module. For thermal fatigue

damage calculation, the maximum and minimumcyclic thermal stresses for the critical points are

taken from the thermal deformation analysis

results. The time life cycles (Nth) corresponding

to each given stress ratio is calculated (Incroperaand David 1990),

log Nf a1 a2 log Smax 1 R n a3 2

R

Smin=Smax; a1

20:68; a2

9:84; a3

0

Smax = the highest algebraic value of stress in thestress cycle. Smin = the lowest algebraic value of

stress in the stress cycle.

The number of cycles (nth) corresponding tothe operation life time of the satellite is calculated

as following:

The satellite duration period (Tdr), which is thetime interval for completing one orbital cycle

(revolution), is defined in the following way:

Equipment A Equipment B

Mounting plan after

deformation

Mounting plan

before deformation

1Avh

nAvh

2Avh

1A'v

h

2A'v

hnA

'vh

1Bvh

2Bvh

nBvh

nB'v

h

1B'v

h

2B'v

h

Fig. 15 The criterion ofangular positioningdeviation calculationbetween any twoequipments


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Tdr 2pffiffiffiffiffile

p a32(

1 eelea

2 1 e2 2"

2 52

sin2i

1 e2 321 e cosu x 2

1 e cosu x 3

1 e2

1 3sin2i sin2u #) 3Where, a = the major semi axis, (= Re + h); Re =

the mean earth radius, (= 6378.14 km); h = the

satellite altitude; le= the gravitational constant ofthe earth, (= 3.986005 105 km3/s2); ee = the

earth oblateness parameter, (= 2.6333 1010 km5/

s2); i = inclination; e = eccentricity; u = theargument of latitude; x = the argument of perigee

For Small Sat, the following parameter values

are taken during calculation, h = 668 km; i =98.085; e = 0.001; u = 0; x = 0. By substituting

with these values into the previous equation, thedraconian period for Small Sat equals to:

Tdr 5881:9 sThe number of cycles (n) corresponding to the

operation life time is defined as:

nth

Topr

Tdr

Fig. 16 Location of the worst thermal case

Table 7 Angular positioning deviation angle for the precise equipments relative to the star sensor

Equipment Angle before deformation,h (deg)

Modified angle afterdeformation, h (deg)

Angular positioning deviationsangle, hdev (arcmin)

Case 1 Case 2 Case 1 Case 2

MBEI 136 135.9871 135.9955 0.777 0.269AVM, gyro Mx 90 89.9982 90.0063 0.105 0.379AVM, gyro My 46 46.0042 46.0117 0.252 0.7002

AVM, gyro Mz 44 43.9885 43.9908 0.691 0.5512AVM, skewed gyro 44 44.0233 44.0238 1.3996 1.428Reaction wheels Mx 90 90.0127 90.0068 0.7614 0.40734Reaction wheels My-1 134 134.0076 133.9993 0.4549 0.0395Reaction wheels My-2 134 134.0035 133.9959 0.2078 0.2446Reaction wheels Mz 44 44.0221 44.0202 1.3242 1.2117Magnetometer 136 135.9947 135.9979 0.3168 0.1239sMagnetorquers 44 43.9828 43.9892 1.0341 0.6458Note:s Case 1 represents the thermal deformation due to on-orbit maximum temperatures.s Case 2 represents the thermal deformation due to on-orbit minimum temperatures.


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Where: Topr is the operation life time of the

satellite, which equals to 5 years.

Topr 5 365 24 60 60 157:86 106 sec

Hence,

nth 26; 808 cyclesThe thermal damage at the critical points is

calculated by the following relation and listed in

Table 8:

Dthermal nthNth

Thermal fatigue damage calculations show

that the satellite structure can withstand allcyclic stresses due to on-orbit cyclic thermal

loading during its life span, because the totalcumulative fatigue damage does not exceed one

for all critical points located in the basis unit

block modules.

10 Conclusion

Satellite structure thermal analysis is done to

verify functional temperature profile, mountingaccuracy of equipments and validate acceptable

thermal fatigue damage. By referring to the

thermal results gained above applying IDEAS,the structure will not experience high tempera-

tures during the mission. Inspection of the aboveresults appears that the maximum temperatures

occur at the onboard computer frame modules

(25.2C). As one can notice from Table 6, there isa difference of 6C between two sides of the first

onboard computer frame module. This tempera-ture difference is due to the temperatures of the

heat shields. The heat shields have a direct impact

on the temperature of the frame modules as fromfacing the heat shield facing the sun or deep

space. As it has been showed in Fig. 7, the

temperature gradient between the plates did notexceed 5C.

Temperature drop in height of reinforcing ribs

was taken equal to 0.2C. For plates this temper-ature increases in the direction of an mounting

plate, and for a case of a basis unitinwards of a

basic unit. In calculation of thermal strains of theplates and basic unit case, a temperature drop in

height of reinforcing ribs was taken into account.

In calculation of cases of frame modules this dropwas not taken into account, since it effects slightly

on angular deviations of instruments requiring anaccurate installation. These temperature distribu-tions are applied in a finite-element model as

boundary conditions and thermal analysis is

executed but using FE technique this time. Thisstep is done to account for the mesh difference

between finite-difference model and the finite-

element one.During on-orbit operation, the satellite is

totally free without any fixation points. To con-

duct a thermoelastic analysis, displacement

boundary conditions must be defined for therelated model. Therefore, the satellite modelmust be constrained by rigid support points.

Selection of these points locations and their

fixation manner should provide minimum effecton satellite thermal deformation. For Small Sat,

the support points are selected at the plane of star

sensor mounting with its bracket. This plane isselected because the mounting accuracy of all

precise equipment installed on the basis unit

Table 8 Thermal and overall fatigue damage calculation for the critical points

Critical points Thermo-elastic analysis Thermal fatigue analysis

rmax rmin N (cycles) n (cycles) Thermal Damage

MPa Ksi MPa Ksi

12 23.575 3.422 27.956 4.058 1.00E+12 26808 2.68E-0813 29.530 4.286 28.697 4.165 1.00E+12 26808 2.68E-0814 1.855 0.269 4.066 0.590 1.00E+16 26808 2.68E-1224 94.158 13.666 125.937 18.278 5.00E+06 26808 5.36E-03


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block is measured relative to the star sensorbracket. Displacement boundary conditions are

applied by fixing one of the star sensor four points

of fixation (on bracket) in the X direction, fixingthe next point in the Y and one of the other points

in the Z direction.

The stress-strain state of the basis unit blockstructure module is determined as a result of the

on-orbit thermoelastic analysis for both worst

cases, in which the maximum and minimumaverage on-orbit surface temperatures take place.

Displacements are relative to the points of star

sensor attachment to its bracket. From staticpoint of view, the basis unit block module is safe

because the yield margin of safety is a positive

value in both design cases. But this is notsufficient to decide whether the design is satisfac-

tory. Satisfactory performance of the satelliterequires accurate prediction of thermal deforma-tions to verify pointing and alignment accuracy

requirements for sensors. The angular positioning

deviation angle hdev (arcmin) is calculated for theprecise equipments mounted on the basis unit

block relative to the star sensor due to on-orbit

thermal deformation. These results are comparedwith the specified limiting angular positioning

deviations for the most precise equipment rela-

tive to the star sensor. Results show that the

performance of the satellite is not affected withon-orbit thermal deformation under maximum or

minimum temperatures.Fatigue damage due to on-orbit cyclic thermal

stresses is evaluated after performing on-orbit

thermal deformation analysis. The maximumthermal stress occurs at the same location under

both worst cases. This location is specified as

Point 24 as shown in Fig. 15 is most severelyaffected by fatigue damage due to on-orbit

thermal cyclic loading. The total fatigue damage

for any point located in the basis unit blockmodule is the dynamic fatigue damage due to

mechanical vibration acting on the satellite duringtransportation and launch plus the thermal

fatigue damage due to on-orbit cyclic thermal

stresses. Therefore, to calculate the overallfatigue damage for the basis unit block, dynamic

fatigue damage must be calculated for Point 30

based on the fatigue analysis of dynamic vibra-tions. Moreover, thermal fatigue damage must be

calculated for the critical points located in thebasis plate and basis walls modules. Thermal

fatigue damage calculations show that the satel-

lite structure can withstand all cyclic stresses dueto on-orbit cyclic thermal loading during its life

span, because the total cumulative fatigue dam-

age does not exceed one for all critical pointslocated in the basis unit block modules. Fatigue

damage due to thermal loading is evaluated and

the satellite structure is safe for the functionalperiod of life time (5 years).

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Thermal Fatigue Analysis