Random Natural Frequencies Research Articles

Wave patterns in frequency-entrained oscillator lattices

We study and classify firing waves in two-dimensional oscillator lattices. To do so, we simulate a pulse-coupled oscillator model aimed to resemble a group of pacemaker cells in the heart. The oscillators are assigned random natural frequencies, and we focus on frequency entrained states. Depending on the initial condition, three types of wave landscapes are seen asymptotically. A concentric landscape contains concentric waves with one or more foci. Spiral landscapes contain one or more spiral waves. A mixed landscape contains both concentric and spiral waves. Mixed landscapes are only seen for moderate coupling strengths g, since for higher g, spiral waves have higher frequency than concentric waves, so that they cannot mix in frequency entrained states. If the initial condition is random, the probability to get a concentric landscape increases with increasing coupling strength g, but decreases with increasing lattice size. The g dependence of the probability enables hysteresis, where the system jumps between the two landscape types as g is continuously changed. For moderate g, spiral tips rotate around a suppressed oscillator that never fires. We call such an oscillator an oscillator defect. A spiral may also rotate around a point defect situated between the oscillators. In that case all oscillators fire at the entrained frequency. For larger g, a spiral tip either moves around a row of suppressed oscillators, a row defect, or around an open curve situated between the oscillators, which may be called a line defect. The length of a row or line defect increases with g. Our results may help understand sinus node reentry, where the natural pacemaker of the heart suddenly shifts to a higher frequency. Some of the observed phenomena seem generic, based on simulations of other models.

Frequency entrainment in long chains of oscillators with random natural frequencies in the weak coupling limit.

We study oscillator chains of the form phi; (k) = omega(k) +K[Gamma( phi(k-1) - phi(k) )+Gamma( phi(k+1) - phi(k) )], where phi(k) epsilon[0,2pi) is the phase of oscillator k. In the thermodynamic limit where the number of oscillators goes to infinity, for suitable choices of Gamma(x), we prove that there is a critical coupling strength K(c), above which a stable frequency-entrained state exists, but below which the probability is zero to have such a state. It is assumed that the natural frequencies are random with finite bandwidth. A crucial condition on Gamma(x) is that it is nonodd, i.e., mid R: Gamma(x)+Gamma(-x) mid R: not equal 0. The interest in the results comes from the fact that any chain of limit-cycle oscillators can be described by equations of the above form in the limits of weak coupling and narrow distribution of natural frequencies.

Phase transitions towards frequency entrainment in large oscillator lattices.

We simulate two-dimensional lattices of pulse-coupled oscillators with random natural frequencies, resembling pacemaker cells in the heart. As coupling increases, this system seems to undergo two phase transitions in the thermodynamic limit. First, the largest cluster of frequency entrained oscillators becomes macroscopic. Second, all oscillators frequency entrain, except possibly some isolated ones. Between the two transitions, the system has features indicating self-organized criticality. To our knowledge, the first transition and the intermediate phase have never been observed before. It remains to be seen if they are generic for large lattices of limit cycle oscillators.

Free vibration of composite cylindrical panels with random material properties

Virtually in all structural systems, and in particular composites, there are uncertainties in the system parameters because of practical bounds on the quality control. In the present work the effect of variations in the mechanical properties of laminated composite cylindrical panels on its natural frequency has been obtained by modeling these as random variables. The transverse shear and rotatory inertia effects have been included in the governing equations. A perturbation approach is presented to obtain the mean and variance of the random natural frequencies. The effects of thickness ratios, edge support conditions and standard deviation of material properties on response of shallow square panels have been investigated. Results have been obtained by employing the finite element method. The approach has been validated by comparison of results with other approaches.

Phase Slips and Phase Synchronization of Coupled Oscillators

The behaviors of coupled oscillators, each of which has periodic motion with random natural frequency in the absence of coupling, are investigated. Some novel collective phenomena are revealed. At the onset of instability of the phase-locking state, simultaneous phase slips of all oscillators and quantized phase shifts in these phase slips are observed. By increasing the coupling, a bifurcation tree from high-dimensional quasiperiodicity to chaos to quasiperiodicity and periodicity is found. Different orders of phase synchronizations of chaotic oscillators and chaotic clusters play the key role for constructing this tree structure.

Free Vibration of a Disordered Periodic Beam

Using a first-order perturbation procedure the statistical properties of the natural frequencies and their associated normal modes of a disordered N-span beam are investigated. Two types of disorder are considered: the random deviation of span length from an ideal design value and the random fluctuation of the bending stiffness along the beam. The grouping pattern of the natural frequencies is used to establish a perfect correlation among certain random natural frequencies; therefore, their relative position can never be reversed. For other natural frequencies a sufficiency condition for nonreversal of position order is given.