Advanced Laboratory for Automation, Robotics and Manufacturing
The Delayed Resonator for Active Vibration Control
Introduction
Research is carried out on the suppression of vibration that may occur as an
undesirable side effect in bridges, machines, aircraft, etc. For suppression
of the vibration, use is made of the novel concept of the Delayed Resonator,
which creates a resonator by feeding back the displacement of the absorber
mass with a time delay, which then resonates at the desired frequency. The
force between the primary and the absorber masses is proportional to the
displacement.
The absorber in effect takes the brunt of the vibrations upon itself.
It is not unlike a pair of good shoes, which protect the feet from rocky
uneven ground, while themselves suffering due to fatigue. The shoes are
expendable, while the feet are definitely not!!! Like the pair of shoes,
absorber merely has to be changed periodically, while the primary mass
remains relatively unharmed.
Delayed Feedback
The amazing thing about the delayed resonator control algorithm is its
simplicity. All that is required is a measurement of the acceleration of
the absorber, which is then fed back with an appropriate delay, and an
appropriate multiplicative gain. The gain and delay values do not depend
upon the primary mass (except for the frequency of vibration), but only
on the properties of the absorber itself.
The scheme is based on the well-known fact that a perfect resonator is
100% effective in suppressing vibrations at its frequency of
resonance.The drawback, however, is that there is no physical
system in the world that is a perfect resonator- all real systems
are dissipative. What we do via the delayed feedback is to destabilize
the system just enough so that it becomes a perfect resonator
in the steady state. In this case, when the system reaches steady state,
the deleterious vibrations are gone! This is of course a theoretical
picture; in reality, all the vibrations will not disappear, if only for
the reason that a computer can achieve only finite precision arithmetic.
But even then, the vibrations are reduced to a large extent, with more
than 40dB of suppression.
What is more, we can create a perfect resonator, not only at one
frequency, but over a range of frequencies. In other words, the delayed
resonator is tunable, and it is tunable in real time.
Recent Developments
Some of the recent foci of active research on the delayed resonator
are listed below.
- Automatic Tuning (Adaptive DR)
A recent development on the DR frontier is the development of an
automatic tuning algorithm. This algorithm is a solution to the problem
that we do not know precisely what the gain and delay should be for a
given disturbance frequency. This is due to our ignorance of the
absorber parameters (a ubiquitous problem), as well as the nonlinearities
in the actuator. The automatic tuning algorithm applies an initial value
of gain and delay, based on a nominal model of the absorber. If
everything were well known, the gain and delay would work perfectly, and
the primary mass would be brought to rest. However, due to the
inaccurate model, it is not. The vibration of the primary structure is
monitored, and a new value of gain and delay is selected based on the
difference between the desired value of the primary acceleration (zero),
and the actual value. These new values of gain and delay are then applied,
and the whole process is repeated. The algorithm converges iteratively to
the true values of the control parameters.
- Dual Frequency DR
Another aspect of the delayed resonator that is being studied is the
possible absorption of two frequencies by using just one feedback
signal (i.e., the absorber acceleration is subjected to one time delay,
multiplied by a gain, and then fed back).
The motivation from this is noticed from the fact that the DR is stable
for some frequencies, and unstable for others. The presence of the time
delay translates into an exponential term (in the Laplace domain) in
addition to the usual polynomial. This equation has infinitely many complex
roots, two of which are placed on the imaginary axis to give the resonance
feature. The other roots, however must all be in the left half of the
complex plane so that their response decays exponentially, and the two
imaginary roots yield the resonance feature in the steady state. This
stability condition cannot always be assured- sometimes the other roots
are in the right half plane, and the system is unstable. This
stability/instability issue depends on the frequency to which the DR is
tuned. Therefore, since the position of the roots depends continuously on
the parameters (gain, delay, etc.), there must be one frequency at which one
pair of the other other roots is on the imaginary axis. That is, not
only does the main pair of roots (those which we have intentionally placed)
on the imaginary axis, but so is another pair of roots! What this
implies is that the absorber structure is resonant at not one, but
TWO frequencies, and hence can absorb both
these frequencies, possibly simultaneously.
At present, the scheme works for a fixed set of (dual) frequencies for
each absorber structure. Research is being carried out to investigate
the absorption of two arbitrary frequencies by a single mass-spring-damper
subsystem.
- Centrifugal Delayed Resonator
The theme of this scheme is to suppress torsional oscillations
using ear-like actively controlled penduli as absorbers. These
penduli are attached to a rotating and oscillating primary body. The absorber
can be tuned to absorb the oscillations of the primary body while leaving the
base rotations unchanged.
Patents
-
Delayed Resonators as Active Dynamic Absorbers (N. Olgac, sole inventor)
US patent no. 5,431,261; issue date: July 11, 1995
-
Single Mass Dual Frequency Fixed Delayed Resonators (N. Olgac, sole inventor)
US patent no. 5,505,282, issue date : April 9, 1996.
-
Tunable Torsional Vibration Absorber: The Centrifugal Delayed Resonator,
(M Hosek, H. Elmali and N. Olgac), filing date : August 10, 1999.
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