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Then, strong and weak interactions of multipliers on the complex plane are studied, and geometric interpretation of these interactions is given. As application of the developed theory the resonance domains for Hill's equation with damping are studied. It is shown that they represent halves of cones in the three-parameter space. Then, parametric resonance of a pendulum with damping and vibrating suspension point following arbitrary periodic law is considered, and the parametric resonance domains are found.

Another important application of damped Hill's equation is connected with the study of stability of periodic motions in non-linear dynamical systems. Then, linear vibrational systems with periodic coefficients depending on three independent parameters: frequency and amplitude of periodic excitation, and damping parameter are considered with the assumption that the last two quantities are small.

Bibliographic Information

Instability of the trivial solution of the system parametric resonance is studied. For arbitrary matrix of periodic excitation and positive definite damping matrix general expressions for domains of the main simple and combination resonances are derived. Two important specific cases of excitation matrix are studied: a symmetric matrix and a stationary matrix multiplied by a scalar periodic function.

It is shown that in both cases the resonance domains is halves of cones ion the three-dimensional space with the boundary surface coefficients depending only on the eigenfrequencies, eigenmodes and system matrices. The obtained relations allow to analyze influence of growing eigenfrequencies and resonance number on resonance domains. Two mechanical problems are considered and solved: Bolotin's problem of dynamic stability of a beam loaded by periodic bending moments, and parametric resonance of a non-uniform column loaded by periodic longitudinal force.

The lecture is a review of the recent results on parametric resonance obtained by the author with Frederick Solem, Pauli Pedersen Denmark , and Alexei A. Mailybaev Russia. Article :. The spectral correlation performs best when the signals at hand have a narrow band power spectrum because in that case the Fourier transform of the convolution between the second power of roll with pitch will be zero most of time except when parametric roll is developing. However, in real sailing conditions the wave spectrum exciting the ship motions can be rather large, and it induces ship responses whose frequency content spans over a wide range of frequencies as well.

Consequently robustification of the spectral correlation is needed.

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This is obtained by bandpass filters that narrow in the roll and pitch signals frequency ranges of interest. The normalization factor in 2. Furthermore, the H0 parameters are estimated on-line. These together served to obtain desired false alarm rates and make the GLRT detector insensitive to changes in sea state. In Sect.

Finally, it is important to point out how the thresholds were chosen. For the GLRT-based detector it was shown that an empirical threshold can be computed based on the estimated H0 distribution of the test quantity g k. After introducing the data sets, the performance of the Weibull GLRT detector is evaluated in both scenarios. Next, the overall robust performance of the monitoring system given by the integration of the spectral correlation detector with the W -GLRT detector is tested.

For the performance assessment of the spectral correlation detector the reader may refer to [14]. The first data set consists of eight experiments run in irregular waves scenario.

The principal dimensions and hydrodynamic coefficients can be found in [18]. The time-history of roll is shown in Fig. Although the vessel experienced parametric roll only once, all the experiments were made to trigger the resonant phenomenon, but in the irregular wave scenario it is somewhat difficult to obtain a fully developed parametric roll resonance, because consecutive wave trains may not fulfill all conditions for its development. This terminology is used in opposition to the regular wave scenario where instead the vessel is excited by a single sinusoidal wave.

Nine hours of navigation were analyzed. Conditions were significant wave height judged by navigators to develop from 5—6 m to 7—10 m. The pitch power spectrum shown in Fig. The model test experimental data set is used to evaluate the capability of the detectors to timely catch the onset of parametric roll; whereas the real navigation data set is used to ensure the insensitivity to usual forced roll. In order to simulate a continuous navigation the single records of the two data set have been stitched together.

Parametric Resonance in Dynamical Systems | naudetife.tk

A smoothing filter was applied around the stitching points to avoid that sudden fictitious variations within the signals at hand could trigger an alarm. Hence the roll time series scrutinized are those shown in Fig. Towing Tank Experiments 20 Exp. Experiment is the only one where parametric roll clearly developed.

Bottom: Roll motion time series recorded during navigation across the North Atlantic Ocean 2. However, a new alarm is suddenly raised when the resonant oscillations take place again. On the sea trial data set the Weibull GLRT detector raises five alarms, which all last for exactly one window length M, as shown in Fig. It is not surprising that a single detector cannot provide full information about the resonance condition since both the phase synchronization and the frequency coupling must be satisfied simultaneously.

Robust detection performance therefore needs simultaneous detection of the presence of both conditions. Visual inspection of the alarm cases suggests that those are false alarms 40 R. To obtain the full picture, the two detectors are combined within a monitoring system, which issues alarms of parametric roll occurrence based upon the tests made by both detectors together. Furthermore, robustness is obtained by making the adaptation to prevailing conditions only when none of the thresholds are exceeded.

This means the H0 statistics and the normalization of the spectral correlation in practice are calculated from data that are older than the data windows used — a few roll periods — and with appropriate forgetting to be able to track changes in weather. The performance of the monitoring system with robustified algorithms is shown in Fig. The general quality of detection performance is apparent. The performance improvement of the Weibull GLRT detector when combined with the spectral correlation detector is shown in Fig. Such data have recently been made available from trials where also wave radar data were logged.

The results with these data were convincing but this validation is outside the scope and space allocated to this chapter. In the spectral domain, spectral analysis provided an indicator for energy flowing from the pitch motion, directly excited by the waves, into roll motion causing resonance. Robustness against usual forced roll motion was shown for both detectors. The detectors showed to be very capable of timely detecting the onset of parametric roll, while achieving a very low false alarm rate.

A necessary part of achieving excellent overall detection performance was obtained by combining the hypotheses from the two detectors. The invaluable collaboration and comments from Dr. Data from model tests were kindly provided by Dr. Storhaug, DNV. Data from the Atlantic passage were logged by M. Blanke in the s and are presented here with the permission from the owner, Maersk Lines. The help received from Mr. Balakrishnan, N. Statistics and Probability Letters, — Basseville, M.

Prentice Hall Belenky, V. Brazil Blanke, M. Bulian, G. Ocean Engineering, — Carmel, S. Journal of the Transportation Research Board, —63 Froude, W.

Transactions of the Institution of Naval Architects, 2: — Transactions of the Institution of Naval Architects, — IFAC, Spain IFAC, Brazil Ginsberg, S. Grimshaw, R. Hashimoto, H.

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Journal of Marine Science and Technology, — Modeling, Identification and Control, — IFAC, Croatia Jensen, J. Springer, Germany Kay, S. Prentice Hall, United States of America Kerwin, J. International ShipBuilding Progress, 2 16 — Levadou, M. McCue, L. Journal of Offshore Mechanics and Arctic Engineering, — Neves, M.

Guedes Soares, Y. Garbatov, and N.

Introduction to Parametric and Autoparametric Resonance

Newman, J. Nielsen, J. In: World Maritime Technology Conference Oh, I. Paulling, J. Journal of Ship Research, 3 1 —46 Shin, Y. Spyrou, K. Umeda, N. Journal of Marine Science and Technology, —23 Umeda et al. In regular waves, it can be evaluated with model experiments or numerical simulation in the time domain within practical accuracy [1—9]. However, this does not simply result in the ability to predict parametric roll in irregular waves because of its practical nonergodicity.

Parametric Resonance

Belenky et al.