An extention of standard results for the acoustic force on an object in a single-mode resonant chamber yields predictions for the onset of oscillational instabilities when objects are levitated or positioned in these chambers. The authors' results are consistent with those of experimental investigators. The present approach accounts for the effects of time delays in the response of a cavity to the motion of an object inside of it. Quantitative features of the instabilities are investigated. The experimental conditions required for sample stability, saturation of sample oscillations, hysteretic effects, and the loss of ability to levitate are discussed

Barmatz, Martin

Rudnick, J.

English

NASA Technical Reports Server

OscillationalInstabilitiesin SingleModeAcousticsLevitators
J. Rudnick
Departmentof Physics,UCLA
405 HilgardAvenue
LosAngeles,CA90024
and
M. Barmatz
Jet PropulsionLaboratory
CaliforniaInstituteof Technology
Pasadena,CA91109
ABSTRACT
An extension of standard results for the acoustic force on an object in a single-mode resonant
chamber yields predictions for the onset of oscillational instabilities when objects are levitated
or positioned in these chambers. Our results are consistent with experimental investigators.
The present approach accounts for the effect of time delays on the response of a cavity to the
motion of an object inside it. Quantitative features of the instabilities are investigated. We
discuss the experimental conditions required for sample stability, saturation of sample
oscillations, hysteretic effects, and the loss of ability to levitate.
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https://ntrs.nasa.gov/search.jsp?R=19910012063 2020-03-19T18:55:27+00:00Z
I wouldlike to discusssomeprogressthat hasbeenmadein the studyof a phenomenon
that is botha technologicalchallengeto the designersof acousticpositionersand an exciting
researchtopic with relevance to fundamentalissues in the mechanicsof solid bodies and
continuousmedia. This phenomenonis illustratedin the behaviorof a samplelevitatedin an
ACESmodulein SpaceShuttleFlightSTS31B,January1984. Thesamplebeganto oscillate,and
those oscillations ultimately resulted in the loss of positioningof the object. While the
oscillationswere initially orderlyand periodic,or at least approximatelyso, they later became
random,or in nonlineardynamicsterminology,chaotic. Oscillationshavealso beenobservedin
ground-based levitators, specifically a triple-axis levitator. Here one sees two kinds of
behavior. Onecan observeoscillationsthatdecayaway. Theyare clearlya transientfeatureof
the motionof the levitatedobject,whicheventuallysettles into a stableposition. This position
representsa "dynamicallystable fixed point." As a secondalternative,there are oscillations
that donot decay. Ratherthey persistat a fixedamplitudefor as longasthe acousticfieldin the
chamber is excited. This kind of oscillation is called a dynamicallystable limit cycle. A
strikingfeatureof this behavioris that it developsspontaneously. That is, the levitated object
can start out essentially stationary, and then it will begin to oscillate with a steadily growing
amplitude until the amplitude of the oscillations saturates at its limiting value.
It is possible to respond to this interesting phenomenon in a variety of ways (Figure 1).
One can regard it as a problem to be overcome by appropriate modifications in the design of the
positioning apparatus -- for example, by utilizing some form of feedback to respond to changes
in the position of the object by changing some aspect of the acoustic drive. Alternatively, one
might attempt to understand the physical mechanism underlying the oscillations and utilize this
knowledge to design more stable cavities, or operate them in such a way as to avoid the
parameter range in which such oscillations occur.
I strongly feel that the superior approach is the one that is based on a clear, physical
understanding of the mechanism for oscillational instabilities. A positioner designed in such a
way as to avoid oscillations will ultimately prove more practical than one that relies on an
466
externalmechanismto detect and damp out oscillationsvia electronicfeedback. I think that
there is an even morefruitful approachto take to this phenomenon.Regardlessof its possible
negative, or even potentially positive, impact on the application of acoustic positioning
technology, the dynamical instability that occurs in acoustic positioner representsa fertile
researchtopic. One stands to gain a significantly improvedunderstandingof a variety of
importantdynamicalphenomena.
in the remainderof this article,I will touchon someof those topics,discussingwhat has
alreadybeenaccomplishedandwhatremainstobedone.
Figure 2 - Adiabatic invariance. The principle of adiabatic invariance as applied to the
single mode levitator allows one to establish a connection between the object's effect on the
resonant frequency of an isolated mode in a high Q cavity and the acoustic forces and torques that
this mode exerts on the object. The principle of adiabatic invariance, as discovered by Boltzman
and Ehrenfest, asserts the following proportionality:
Facoustic _ _Z_o°_(7o)
Here _ is the resonant frequency of the cavity and r0 is the position of the object of interest. The
constant of proportionality can be calculated in a variety of interesting cases (Figure 3). It
depends on quantities like the intensity of the acoustic field. This relationship has been derived
and it has been tested experimentally. It provides a useful way of calibrating single mode
levitators.
Figures 4 and 5 - The limits of adiabatic invariance. The above relationship applies
when all changes in the position and orientation of the object in the chamber are quasistatic.
Furthermore, it assumes that the acoustic mode is both undriven and undamped. For example,
assume that we are in the real world in which acoustic modes are damped and must be driven.
What if changes that occur in a system occur at a finite rather than infinitesimal rate? In the
case of an acoustic positioner, we have the beginning of an answer to those questions. Garrett
467
and Barmatzhave shownthat in an analoguesystem(thedamped,drivenharmonicoscillator),
parametric changes occurring at a finite rate can lead to instabilities that are strikingly
similar to the oscillationalinstabilitiesthat are observedin acousticpositioners. Dr. Barmatz
and I have verified the results of Garrett and Barmatz by performinga highly nonlinear
calculation of the forces on a small spherical object that is moving inside a single mode
chamber. We have begun to explore in a greater generalitythe effects of time delays and
mechanicalfeedbackon the motionof an objectof arbitraryshapeand size in a singlemode
chamber. Here we are investigatingvirgin territory in dynamics,in that we are attemptingto
understandin a systematicway the limitsof adiabaticinvarianceand the novelconsequencesof
the corrections to that important dynamical principle that must be introduced to describe the
behavior of real, physical objects.
As an example of the progress achieved so far, Figure 6 shows curves that we have
calculated for the threshold and saturated amplitudes of oscillations in a single mode acoustic
levitator operating in a gravitation-free environment. The amplitudes normalized to the
dimension, L z, of the chamber are plotted against the frequency of the drive minus the resonant
frequency of the chamber, normalized to the half-width of the resonance.
There are other modifications to adiabatic invariance that merit study (Figure 7). The
investigations described above have been, or will be, carried out on cavities in which the
relevant acoustic mode is isolated. The question of what happens when the positioning is
accomplished by the excitation of degenerate, or nearly degenerate, modes also deserves our
attention. The triple axis levitator utilizes three exactly degenerate plane wave modes in order
to levitate and position the object inside. This levitator can rotate the object inside at a
controllable rate. A triple-axis levitator was used in the Drop Dynamics Module (DDM) on
Shuttle Flight. The rate of rotation is controlled by the relative phase of two of the plane wave
modes. There is, as yet, no entirely satisfactory explanation for this phenomenon. A proper
description will involve a detailed discussion of the energy exchange that occurs between two
degenerate modes, most probably in violation of the principle of adiabatic invariance. It is
468
interestingthat a recent issueof Physical Review Letters contains an article describing anti-
adiabatic behavior in a quantum mechanical system at the point of level crossing (or
degeneracy).
Figure 8. Another aspect of the oscillational instability that deserves study is the
possibility of chaotic motion by the moving object. Recall that the object in the videotaped ACES
module executed motion that was far from regular. It is altogether likely that this motion fits
the now-accepted definition of chaos -- in that it is locally deterministic but unpredictable in
the long run. However, any attempt to understand that motion will be complicated by the fact
that one must take into account the fact that the acoustic drive was constantly being
electronically adjusted to home-in on the resonant frequency of the cavity. This "external"
feedback gives rise to a more complicated set of equations. We are currently simulating the
motion of a small sphere in a single mode cavity using equations that properly describe the
motion of the atmosphere in a cavity subject to a fixed-acoustic drive. We hope to be able to
construct a full dynamical phase diagram of the object's motion, and, in the process, to explore
the interplay between chaotic motion of a solid object and the concomitant behavior of the
medium whose acoustic field is levitating it.
Figure 9. Finally, it is worth pointing out why it is a good idea to perform experiments
on the motion of levitated object in space rather than in an earth-bound laboratory. The
microgravity environment possesses three distinct advantages. First, the relatively low
acoustic intensities required to position an object in microgravity (as opposed to the
155-160 dB that is needed in 1 g) make it possible to eliminate unwanted effects, such as
rotational instabilities, and also to control nonlinearities in the system. Second, it is possible
to utilize a less dense levitating medium in microgravity, so the system becomes more nearly
Hamiltonian, because of the reduced viscous drag. Finally, in microgravity it is possible to
position a sample at a velocity antinode and exploit the full symmetry of the levitating system.
469
FIGURE 1 TWO ASPECTS OF OSCILLATIONAL INSTABILITIES
TECHNOLOGICAL
HOW DOES ONE AVOID OR EXPLOIT INSTABILITIES?
REACTIVE APPROACH
DESIGN THE ACOUSTIC DRIVE WITH FEEDBACK THAT
ALLOWS IT TO RESPOND TO EVENTS IN THE
POSITIONER
BASIC SCIENCE
• APPROACH: UNDERSTAND THE MECHANISMS LEADING TO
INSTABILITY
• DESIGN AND OPERATE THE POSITIONER IN ACCORD WITH
THE INSIGHT GAINED
470
EJ_GL[_P:_.__ IMPLICATIONS OF OSCILLATIONAL INSTABILITIES
ADIABATIC INVARIANCE
BOLTZMANN - EHRENFEST PRINCIPLE
APPLIES WHEN
THE OBJECT IS STILL OR MOVING ASYMPTOTICALLY
SLOWLY ("QUASISTATIC")
THE POSITIONING MODE IS ISOLATED (NO ENERGY
EXCHANGE WITH NEARBY ACOUSTIC MODES)
DELAY EFFECTS ARE UNIMPORTANT
• THE POSITIONING MODE IS UNDAMPED AND UNDRIVEN
471
FIGURE 3
Jl J
- object
u (eo,O,s_,s2....)
POSITIONING POTENTIAL
_0 = POSITION OF OBJECT
O = ORIENTATIONOFOBJECT
ACOo(?o,O sl....)
PERTURBATION OF THE MODE'S
NATURAL FREQUENCY
POSITIONING FORCE
S1..... SHAPE PARAMETERS (CORRESPONDING RELATIONSHIPS FOR SHAPING
FORCES)
GENERALIZES RESULTS OF KING & GOR'KOV
ALLOWS FOR CALIBRATION OF LEVITATING MODULES
472
FIGURE 4 THE LIMITS OF ADIABATIC INVARIANCE
SUPPOSE THE MODE IS DAMPED & DRIVEN DELAY EFFECTS ARE NON-
NEGLIGIBLE AND THE POSITIONED OBJECT IS MOVING AT A FINITE
RATE.
WHAT HAPPENS THEN?
• S. GARRETTAND M. BARMATZ:
n_d2 + "t'dx + k (t)x = Fe i_t- Fe -i°x
• dt 2 dt
RATE AT WHICH WORK MUST BE PERFORMED TO CHANGE THE
SPRING CONSTANT, k (t), is
1 _x2)dk
• =2-" "dt
473
FIGURE 5
SOLVING FOR x (0
! (x(t)2)_-I - 1 FO 2 1 dk
2 2 4_2m2 / 7 )2 dt
_m + (_'_ -03)2
Resonant frequency of the oscillator
+
4F0 2
[2 m]
(_-{o)
2m [dkt2
[{2__) + (n _{0)213 'dt/
A velocity
dependent force
ENERGY IS FED INTO THE SYSTEM IF co< f_
ENERGY IS EXTRACTED FROM THE SYSTEM IF co< f_
J. RUDNICK AND M. BARMATZ
VERIFIED BY EXPLICIT CALCULATION THAT THIS APPROACH HOLDS FOR
SMALL SPHERES IN A LEVITATING OR POSITIONING CHAMBER
474
FIGURE 6
0.6
0.4
8
02
.J
o
0.0
I I I I
g=O
! ! ! I
0.0 0.1 02. 0.3 0.4
FREQUENCY SHIFT (e - _ ((}))If
0.5
475
FIGURE 7
UNANSWERED QUESTIONS:
• DOES THE APPROACH OF GARRETT AND BARMATZ HOLD FOR AN
ARBITRARILY SHAPED SAMPLE OF ARBITRARY SIZE?
CAN ONE PRODUCE A GENERALIZED PRINCIPLE LIKE THE ONE
FOLLOWING FROM BOLTZMANN-EHRENFEST ADIABATIC INVARIANCE
TO DESCRIBE OSCILLATIONAL, ROTATIONAL, AND DISTORTIONAL
INSTABILITIES FOR LEVITATED OBJECTS?
• WHAT HAPPENS WHEN THERE ARE TWO (OR MORE) DEGENERATE
MODES
• TRIPLE AXIS LEVlTATOR
• SOURCE OF ROTATIONAL CONTROL
• ANTI ADIABATICITY
476
FIGURE 8
MOTION OF A LEVITATED (OR POSITIONED) OBJECT
• FIXED POINT (STABLE LEVITATION)
LIMIT CYCLE (STABILIZED OSCILLATION)
CHAOS (???)
INTERPLAY BETWEEN MOTION OF A COMPACT OBJECT AND THE
MODE(S) THAT SUPPORT IT
477
FIGURE 9 WHY WORK IN SPACE
INTENSITY OF POSITIONING FIELD CAN BE REDUCED.
155-160dB THE MINIMUM REQUIRED AT GROUND LEVEL.
UNDESIRABLE EFFECTS ARE ELIMINATED (I.E., ROTATIONAL
INSTABILITIES).
ENVIRONMENT CAN BE MORE FULLY CONTROLLED (LESS DENSE
MEDIUM - LESS DRAG)
FULL SYMMETRY OF THE CAVITY CAN BE EXPLOITED (OBJECTS
ARE POSITIONED AT CENTER)
478


Oscillational instabilities in single mode acoustics levitators

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