Qualification Testing Phase of Satellite Structure - Finite Element Analysis for Satellite Structures - page 265

Table 6.4 Natural
frequencies of the
satellite [ 4 ]

Frequency Damping

43 0.083

50 0.081

80 0.07

100 0.076

r 0 ¼ r j o þ Q 1 1 e be pl

ð 6 : 3 Þ

where Q 1 and b are material parameters. Parameter Q 1 is the maximum change
in the size of the yield surface, and b defines the rate at which the size of the yield
surface changes as plastic straining develops [ 5 ]. [156 MPa, b = 5.5].
The most dangerous (from the point of view of vibration strength) vibrations
are with a frequency not more than 100 Hz. Therefore, in calculations the vibra-
tions are taken into account only in the frequency band of 100 Hz. At first, modal
analysis was carried out. Table 6.4 shows the natural frequencies—the result of
modal analysis and damping ratio—the value used in harmonic analysis. For all
loading cases the satellite structural analysis is performed with a consideration of
the action of sinusoidal vibrations at each resonance frequency f i . In the analysis a
damping value is given, which equals [ 6 ],

n ¼ 1 =ð 10 þ 0 : 05 f Þ ð 6 : 4 Þ

Harmonic analysis is performed to calculate peak stresses in the finite element
model of the basis-unit subassembly and the virtual satellite structure due to har-
monic loads [ 6 ]. The output results from the harmonic analysis will be used in the
fatigue analysis. The harmonic tension stresses are responsible for dynamic fatigue
failure. Harmonic loads do not act individually on structural elements, but they
operate simultaneously with quasi-static loads. Quasi-static loads comprise both
static and dynamic loads, and are applied at a frequency sufficiently below the first
natural frequency of the structure. Therefore, the quasi-static loads are independent
of time or vary slowly, so that the dynamic response of the structure due to the
dynamic component is not significant. The maximum stresses affecting the structure
are a combination of the equivalent amplitude stresses due to fully reversed har-
monic loads and the equivalent stresses from the static components of the quasi-
static loads. Since the finite element model of Small Sat is linear, superposition is
applicable to calculate the maximum stresses in each structural module.
When the cyclic load level varies during the fatigue process, a cumulative
damage model is often hypothesized. The generalization of this approach is called
Miner's Law and can be written

X n j
N j

¼ D ð 6 : 5 Þ