# Elah Bozorg-Grayeli Vahe Gabuchian Zac Lizer Francisco Montero Chacon Elliott Pallett Matt Wierman NuSTAR Structural/Thermal Report

Urban and Civil

Nov 15, 2013 (4 years and 6 months ago)

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Elah Bozorg
-
Grayeli

Vahe Gabuchian

Zac Lizer

Francisco Montero Chacon

Elliott Pallett

Matt Wierman

NuSTAR Structural/Thermal Report

The purpose of the Ae 105c NuSTAR project was the development and validation of
accurate structural, dynamical, and metrolog
y models and systems. In order to accomplish this,
the class divided into three teams to conduct their analysis. This report covers the results
obtained by the Structural/Thermal team, whose assignment was to create a robust structural
-
thermal integrated m
odel of the boom.

Before beginning computational
modeling of the discrete boom elements
, we decided to
model the structure using Euler
-
Bernoulli beam theory. We considered the cases for bending,
axial extension, and torsion. Within these cases, we used mul
tiple boundary conditions in order
to obtain as much

frequency

data on the boom as possible. We analyzed the cases in which:

1.

One end is clamped, one end is free, no mass on the ends

2.

One end is clamped, one end is free, with a 100 kg mass on the free end

3.

Bo
th ends are free,
200 kg mass on one end

4.

Both ends are free, with 200 kg on
both

end
s

Having decided to consider these cases, we needed to choose how to model the beam. Since
the cross
-
sectional area of this structure changes along the length of the boom,

and since there
are additional masses at each of the joints along the beam, modeling the structure as a continuum
is extremely difficult. Accordingly, we decided to model the boom as
four

beams, one at each
corner of a square of 0.233 m width. Each boom h
ad a 0.01 m diameter and a length of 10.91 m.
The structural properties of the
beam were those of P75 Graphite Epoxy
1
:

Young’s Modulus

=
170 GPa

Density = 1.6 g/cm
3

Poisson’s Ratio

assumed to be .3

Given these properties, we used the Euler
-
Bernoulli
e
quations shown in Appendix A

to
calculate the natural frequencies of the beam.
Provided below is a table of s
ome of the values we
calculated:

After completing cases 1 and 2, and checking with the
initial
FEM code, we determined
that the difference
betwe
en the calculated natural frequencies

was too high. Therefore, a
continuum model of the truss was considered insufficient for our purposes.

Two software packages were then used to generate FEM models of the boom. JPL
provided data for the boom configuratio
n: dimensions and material properties. Models were
generated in ANSYS and CALFEM and compared to data provided by the boom’s manufacturer.
The data on modal frequencies was calculated using a cantilever
-
free configuration where the
beam is attached to a fi
xed location and is allowed to vibrate freely. No additional masses are
considered. The FEM models we generated assumed the structure was pin
-
jointed and structural
properties were as defined by the manufacturer.

The oscillations of a structure such as th
is are governed by the equation:

2
2
0
d x
M Kx
dt
 

W
here K is the structure’s stiffness matrix and M is the structure’s mass matrix. The eigenvalues
correspond to the frequencies of the modes (which can be converted to Hz) and the eigenvectors
v
isually represent the modes.

Our results for the first analysis are shown in this table:

Case 1

CALFEM (Hz)

ANSYS (Hz)

Bending Mode 1

2.97

3.41

Bending Mode 2

15.39

17.61

Bending Mode 3

35.41

40.52

Axial Mode

130.61

149.38

Torsional Mode

11.13

12.60

Our goal to model flight conditions drove us to add a 100kg mass to the end of the
cantilever beam. With no test data to compare to, we compared the results given by each
software package. The only difference between the two modes is the
modeling of the cross
-
cables for each bay. CALFEM lacks the functionality required to have members as cables (it
only allows bars and beams). The ANSYS analog is a model with pre
-
stressed cables. The
degree of pre
-
stress applied is determined by the point
at which the structure will deflect (within
reasonable limits) without any cables becoming slack. The results are self
-
consistent:

Case 2

CALFEM (Hz)

ANSYS (Hz)

Bending Mode 1

0.62

0.63

Bending Mode 2

11.25

12.83

Bending Mode 3

29.27

33.38

A
xial Mode

34.23

34.57

Torsional Mode

2.71

2.73

Having completed this case, we continued our efforts to better model flight conditions.
Since the cantilever configuration is an over simplification of the problem, we chose to model
the beam as an un
constrained element with a mass at each end. One end has 100kg to match the
optics and the other has a 200kg mass to match the S/C bus. These two masses are of the same
order of magnitude and thus the cantilever configuration
is

not realistic. For the free
-
free
configuration, we kept the assumptions of a pin
-
jointed structure, and continued with the same
FEM models, with the same setup as before. Note: the ANSYS and CALFEM models still differ
in the cable modeling approach. As you can see below, both models

produced results for the
modes of bending, torsion and elongation that were comparable:

Case 3

CALFEM (Hz)

ANSYS (Hz)

Bending Mode 1

8.19

9.28

Bending Mode 2

26.19

29.63

Bending Mode 3

47.02

52.63

Axial Mode

42.66

42.82

Torsional Mode

3.36

3.38

For similar frequencies as those found in bending/elongation/torsion, we found another
mode we will call ‘parallelogramming.’ Since in both FEM models we have no way to constrain
the ends of the boom into a square configuration, this mode ex
ists. This does not parallel the real
case, as the S/C bus and optics bus would constrain this movement at their attachment points.

The final step of our modal analysis sought to cover the last major discrepancy between our
model and the real boom. The lo
ngeron/batten joints can sustain moments in their deployed
configuration, whereas we had been assuming a pin
-
jointed structure for our previous analyses.
Due to difficulties in modeling this moment
-
carrying configuration in CALFEM, we proceeded
with the AN
SYS model alone. The previous model was altered to allow moments to be
transferred between the battens and longerons but not the pre
-
stressed cables.
The results shown
here represent our best approximation of the flight hardware configuration:

Case 4

ANSYS

(Hz)

Bending Mode 1

1.4

Bending Mode 2

15.32

Bending Mode 3

33.56

Axial Mode

42.69

Torsional Mode

24.24

Having completed the structural analysis of the boom, we began analyzing the reaction to
thermal loads. Initially, analytical work wa
s carried out to determine the temperature of a single
member in constant sun and in constant shade. These steady
-
state temperatures could provide
bounds for the later numerical measurements; the maximum and minimum cyclic temperatures
should not go beyon
d these numbers.

To begin modeling multiple members and varying heat loads, a simplifying assumption
had to be made; namely, that the individual members were isothermal. To confirm this
assumption, a FD model was written in MATLAB that broke a single mem
ber longitudinally into
three pieces and subjected it to cyclical thermal loading designed to replicate the loading
experienced by the boom in orbit. Allowing for conduction between the pieces, the temperature
was found to vary less than 1 degree Kelvin o
ver the width of the boom, verifying the isothermal
member assumption.

With this completed, the truss could be modeled in full. A model was created that
calculated the temperature of the 12 members (8 battens, 4 longerons) that made up a bay of the
boom.

Documents from JPL revealed that conduction between members was essentially
nonexistent, so only radiative coupling had to be considered. The view factors between the
members were calculated using formulae from this website
(
http://www.me.utexas.edu/~howell/tablecon.html#C3
). The symmetry of the boom was
exploited by only modeling a single bay, as radiation between bays should be negligible and each
bay is subject to the same loading. T
his early model could calculate the temperature of the
members for several orbits with the long axis of the boom normal to the incoming solar
radiation. A case was considered where two longerons completely shaded the other two
longerons, this having been
determined to be the
worst
-
case

thermal coatings were tested and white paint was found to produce the smallest temperature
gradient between the insolated and shaded members.

Following PDR the model was expanded to inclu
de the effects of tilt away from the sun
(both the decreased heat flux and possible shading by the spacecraft bus).
, the
temperature of the aluminum joints connecting the members was calculated, as the high CTE of
aluminum means these are impo
rtant to the distortion of
the boom. Finally, a more reasonable loading case was
considered. In this example, the boom was tilted slightly
(10º) away from the sun, with the bus on the shade side
so that it did not shade the mast. Three longerons and all

the battens were fully in sun, with only intermittent
shading of the fourth longeron (see figure, red =
insolation, black = shade). All the joints were also in
sun. The case is representative of the vast majority of orientations the spacecraft will assu
me
while in orbit. Using CTE’s from the ADAM boom documents for P75 graphite/epoxy members
and aluminum joints, the distortion caused by the temperature gradient across the boom was
found to be well within the operational parameters of the sat
ellite.

Th
is structural and thermal analysis has demonstrated that the NuSTAR boom meets the
JPL level 1 requirements placed upon it. Further information about the boom, including the FEM
and FD codes are available at
http://ae105.wikispaces.com/Structures+and+Thermal
.

References

1.

Kegg, Colleen M., “
Thermally
-
Stable Deployable Structure
.” Goleta: AEC
-
ABLE
Engineering Company, Inc.

2.

Blevins, Robert D.,
Formula for Natural Frequency and Mode Shape
.

New York: Van
Nostrand Reinhold, 1979

3.

Nestorides, E.J.,
A Handbook on Torsional Vibration
. Cambridge: Cambridge University
Press, 1958