Periodic structures can effectively suppress vibrations through the use of stop bands. Stop bands are frequency ranges where vibrations are unable to propagate through a periodic structure. Two types of periodic structures are phononic crystals and locally resonant structures. Phononic crystals use a repeating pattern of multiple materials to generate stop bands. Mathematical models and simulations show that a phononic crystal structure is able to attenuate vibrations within stop band frequencies, while a uniform structure does not. Locally resonant structures contain internal resonators that interact with a base structure to also produce stop bands. Both types of periodic structures have potential to isolate delicate spacecraft components from vibrations.

1. Vibration Suppression Via Periodic Structures
Chris Gnam and Mostafa Nouh, PhD.
The State University of New York at Buffalo, Department of Mechanical and Aerospace Engineering
Abstract
Spacecraft optical equipment often times needs to be held
extremely still while taking long exposures, and so they must
somehow be kept isolated from undesirable sources of vibration
(such as from reaction wheels). Materials with good damping
properties, such as rubber-like and viscoelastic polymers, are
typically soft with a low mechanical stiffness and are thus not
always suitable. Another approach is to use periodic structures
(utilizing both periodic geometries and materials) which are
known to exhibit a unique structural response stemming from
their ability to generate stop bands. Within these stop bands,
vibration excitations of certain frequencies are incapable of
producing elastic waves that propagate through the structure
and are effectively damped. Here we study periodic structures
using ANSYS workbench to model their filtering characteristics.
We have also created mathematical models using MATLAB to
predict the wave dispersion patterns of these structures which
have allowed us to identify geometries with potentially powerful
vibrational suppression properties, which may allow for more
precise data collection on spacecraft optical systems.
Introduction
SORA Optical System
Motivation
Methods
Frequency Response Frequency Response &
Negative Effective Mass Concept
Conclusions
Acknowledgements
References
(1) Phononic Crystals
• Vibrations can be extremely detrimental in space
applications.
• Delicate components can be damaged during launch, and
vibration sources while in space can wreak havoc on optical
equipment or sensitive sensors on spacecraft.
• Sensitive components should be kept isolate from potentially
“dangerous” vibrations.
• Historically many applications have used extremely complex
structures, or power consuming actuators (such as piezo
electrics) for an active solution.
• Active methods are reliable and have been employed in
spacecraft such as the Hubble Space Telescope and the
Rosetta spacecraft at 67P/Churyumov-Gerasimenko, but
they are also expensive, power consuming and complex to
manufacture within the spacecraft’s structure.
• A different solution might be the use of periodic structures.
These offer a passive (non-energy consuming), lightweight,
and mechanically simple solution. They can be easily
integrated into a spacecraft's structure and can provide
some amount of vibration control to help isolate sensitive
components.
Even simple periodic structures can provide decent vibration
suppression. By combining various different geometries, materials
and structures together, we were able to suppress vibrations
through materials. More complex structures, and more carefully
selected materials might be able to do even better. This type of
technology would be invaluable in the development of Spacecraft
Optical systems.
The University at Buffalo Nanosatellite Laboratory is constructing
a satellite which aims to characterize space debris via
spectroscopy.
This requires not only an incredibly precise Attitude Control
system, but also it requires the Spectrometer remain isolated
from all potential sources of vibrations.
• Periodic structures have a unique filtering behavior that stems
from their ability to exhibit stop bands, i.e. ranges of frequencies
within which external vibration excitations are incapable of
infiltrating the structural medium.
• There are 2 fundamental types of periodic structures that can
generate a stop band behavior:
1) Phononic Crystals (PC)
2) Locally Resonant Structures (LRS)
• Derive the governing equations of motion of a phononic crystal
and compare with a uniform structure of equal geometry.
• Plot the response of both structures to an external excitation at
different frequencies.
• Derive the transfer matrix of the PC unit cell and use MATLAB to
find its eigenvalues which allows us to predict frequencies
corresponding to stop bands.
• Compare to ensure stop bands are where we expect them to be.
• Dr. Mostafa Nouh, Assist. Professor and Laboratory Mentor
• For the financial and training support of CSTEP
• University at Buffalo Nanosatellite Laboratory
Fig. 1 – SORA Optical System
Table 1 – Dimensions and material properties of a 1D phononic crystal
Fig. 2 – A cantilevered 1D phononic crystal of a 2-material unit
cell and a total of 10 cells
Fig. 3 – Frequency response of phononic vs uniform Rod
• Phononic crystals, are composite-like structures that are made
up of multiple materials which are placed in some kind of a
self-repeating pattern.
• After defining the PC’s unit cell, it is possible to derive a
transfer matrix (which describes a wave’s movement from one
cell to the next). Transfer matrices can help us mathematically
predict the frequency locations of stop bands.
Results & Conclusions
• The wave attenuation constant α can be computed from the
eigenvalues of the transfer matrix of cell k shown in Fig. 2.
• When α is zero, this indicates the wave is freely propagating
through the structure with no attenuation. When it’s equal to a
non-zero value, the wave is attenuated and eventually blocked
from propagating. Therefore, the response of the free-end of
the PC shows a stop band as shown in Fig. 3.
• This is confirmed by the ANSYS simulation shown in Fig. 4. In
the Uniform rod, vibrations are able to pass through freely
causing deformations throughout. In the periodic rod,
vibrations are attenuated and so deformations only occur at the
point of excitation and die off quickly. (These results were
obtained at 4.5 kHz, which is a stop band for the Phononic Rod
as indicated by the Frequency Response curves)
Material Young’s Modulus
(GN/m2)
Density
(kg/m3)
Cross Section
(m2)
Length/cell
(m)
A 210 7,800 0.000625 0.025
B 0.025 1,200 0.000625 0.025
AL
A B
unit cell
cell k − 1 cell k + 1cell k
,A Ru
,A RF ,B LF
,B Lu
,A LF
,A Lu ,B Ru
,B RF
 
BL
(2) Locally Resonant Structures
• Locally resonant structures are structures which are equipped
with internal sources of mechanical resonance.
• The interaction of the internal resonators with the outer (base)
structure gives rise to another class of frequency stop bands.
• Stop bands in LRS are computed using the dispersion
curves which give the relation between the exciting frequency
and the wavenumber of the propagating waves, if any (Fig. 6).
Fig. 5 – A simple locally resonant structure (mass-in-mass system)
Fig. 6 – Dispersion curves of a locally resonant structure.
Fig. 7 (a) Frequency response of the LRS showing the stop band region. (b) A
structure without internal resonators can only achieve a stop band in the same
frequency range if its mass is negative, which is not physically possible.
• The frequency response (obtained from the equations of
motion and using MATLAB) shows a stop band in the same
region predicted by the dispersion curves.
• We also plotted the effective mass of a virtual structure without
internal resonators and have shown it can only achieve a stop
band in the same frequency range if its mass is negative,
which is not physically possible. In the metamaterials field, this
is referred to as the negative effective mass concept.
Phononic Rod
(Materials A and B)
ANSYS Simulation
Uniform Rod
(Material A only)
Deformation at 4.5 kHz
Fig. 4 – Deformation of the phononic vs. uniform rod inside a stop band
frequency using ANSYS WorkBench
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