The Delayed Resonator for Active Vibration Control

This project involves a novel vibration absorption methodology. It is commonly known that a resonator (i.e., a mass attached to a perfect spring) can absorb vibrations entirely at its natural frequency. Consider having a vibrating system with a stable (i.e. dissipative) absorber attached to it. Because of the dissipative characteristic (i.e. damping component) this absorber cannot suppress oscillations entirely. We wish to convert the absorber into a resonator, and we also want to moderate the resonance frequency of it. And how would we achieve that? Simply by introducing a time delay into the feedback loop! Sounds simple, doesn’t it? It is, indeed, once the proposition is well understood. Especially the “delay in feedback”, which is considered to be a destabilizing effect, bring some interesting challenges. Among these are: multiple frequency resonances, stability of delayed systems, multiple delay involved systems stability. Please see our selected publications for the treatment of these topics.

This project was featured in CNN Science and Technology Week and  in WTNH, CT Channel 8 news.

CNN Science and Technology Week, 1994

WTNH, CT Channel 8 News, 1994


  • 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. The following video shows the absorber working on a steel plate. Please click on the picture to watch it.

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.

This project led to three patents, which are presented in this link.