Visualization of Escher-like Spiral Patterns in Hyperbolic Space

Spirals, tilings, and hyperbolic geometry are important mathematical topics with outstanding aesthetic elements. Nonetheless, research on their aesthetic visualization is extremely limited. In this paper, we give a simple method for creating Escher-like hyperbolic spiral patterns. To this end, we first present a fast algorithm to construct Euclidean spiral tilings with cyclic symmetry. Then, based on a one-to-one mapping between Euclidean and hyperbolic spaces, we establish two simple approaches for constructing spiral tilings in hyperbolic models. Finally, we use wallpaper templates to render such tilings, which results in the desired Escher-like hyperbolic spiral patterns. The method proposed is able to generate a great variety of visually appealing patterns.

The Effects of Symmetry Breaking Perturbation on the Dynamics of a Novel Chaotic System with Cyclic Symmetry: Theoretical Analysis and Circuit Realization

Symmetry is an important property shared by a large number of nonlinear dynamical systems. Although the study of nonlinear systems with a symmetry property is very well documented, the literature has no sufficient investigation on the important issues concerning the behavior of such systems when their symmetry is broken or altered. In this work, we introduce a novel autonomous 3D system with cyclic symmetry having a piecewise quadratic nonlinearity [Formula: see text] where parameter [Formula: see text] is fixed and parameter [Formula: see text] controls the symmetry and the nonlinearity of the model. Obviously, for [Formula: see text] the system presents both cyclic and inversion symmetries while the inversion symmetry is explicitly broken for [Formula: see text]. We consider in detail the dynamics of the new system for both two regimes of operation by using classical nonlinear analysis tools (e.g. bifurcation diagrams, plots of largest Lyapunov exponents, phase space trajectory plots, etc.). Several nonlinear patterns are reported such as period doubling, periodic windows, parallel bifurcation branches, hysteresis, transient chaos, and the coexistence of multiple attractors of different topologies as well. One of the most gratifying features of the new system introduced in this work is the existence of several parameter ranges for which up to twelve disconnected periodic and chaotic attractors coexist. This latter feature is rarely reported, at least for a simple system like the one discussed in this work. An electronic analog device of the new cyclic system is designed and implemented in PSpice. A very good agreement is observed between PSpice simulation and the theoretical results.

Generation of advanced Escher-like spiral tessellations

In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give a way to achieve simple spiral tilings of cyclic symmetry. Then, we introduce several conformal mappings to generate three derived spiral tilings. To obtain Escher-like tessellations on the generated tilings, given pre-designed wallpaper motifs, we specify the tessellations’ implementation details. Finally, we exhibit a rich gallery of the generated Escher-like tessellations. According to the proposed method, one can produce a great variety of exotic Escher-like tessellations that have both good aesthetic value and commercial potential.

Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks

The paper describes a numerical method based on a modal work approach to evaluate the forced response of bladed disks and its validation against numerical results obtained by a commercial FEM code. Forcing functions caused by rotor–stator interactions are extracted from CFD unsteady solutions properly decomposed in time and space to separate the spinning perturbation acting on the bladed disk in a cyclic environment. The method was firstly applied on a dummy test case with cyclic symmetry where the forcing function distributions were arbitrarily selected: comparisons for resonance and out of resonance conditions revealed an excellent agreement between the two numerical methods. Finally, the validation was extended to a more realistic test case representative of a low-pressure turbine bladed rotor subjected to the wakes of two upstream rows: an IGV with low blade count and a stator row. The results show a good agreement and suggest computing the forced response problem on the finer CFD blade surface grid to achieve a better accuracy. The successful validation of the method, closely linked to the CFD environment, creates the opportunity to include the tool in an integrated multi-objective procedure able to account for aeromechanical aspects.

Type II Vacuum Spacetime Admitting Closed Timelike Curves

We present a cyclic symmetry type II vacuum spacetime admitting closed timelike curves (CTCs) which appear after a certain instant of time, i.e., a time-machine spacetime. The various authors in past have considered the 2D and 4D flat generalization of Misner space, but in the present work, we have considered the curved spacetime generalzations of 4D Misner space, and is asymptotically flat radially

The Effects of Gravity on the Response of Centrifugal Pendulum Vibration Absorbers

This paper describes the effects of gravity on the response of systems of identical, cyclically arranged, centrifugal pendulum vibration absorbers (CPVAs). CPVAs are passive devices composed of movable masses suspended on a rotor, suspended such that they reduce torsional vibrations at a given engine order. These absorbers are becoming prevalent in automotive powertrain components in order to expand fuel-efficient engine operating conditions. Gravitational effects acting on the absorbers can be important for a horizontal rotor/CPVA system spinning at relatively low rotation speeds, for example, during engine idle conditions. The main goal of this investigation is to predict the response of a CPVA/rotor system in the presence of gravity. A linearized model which includes the effects of gravity and an order n torque acting on the rotor is analyzed by exploiting the cyclic symmetry of the system. The results show that the N absorbers respond in one or more groups, where the absorbers in each group respond with identical waveforms but shifted phases. The number of groups depends on the engine order n and the ratio Nn. It is shown that there are special resonant effects if the engine order is n = 1 or n = 2, the latter of which is particularly important in applications. In addition, it is shown that for N > 1 the rotor response is not affected by gravity, due to the symmetry of the gravity effects. The analytical predictions are verified by direct simulations of the equations of motion.

Active detection of small imperfections in structures with cyclic symmetry

Structures possessing cyclic symmetry such as turbine bladed disks, ultrasonic motors, and toothed gear wheels can experience elevated vibration levels when small deviations from circumferential periodicity exist. Detection of these perturbations via classical system identification approaches is time-consuming, indirect, and exhibits low sensitivity to defects and are affected by measurement noise. The present work utilizes low-level forces that automatically lock onto a weighted rotating projection of the system modes at resonance frequency to enhance the detectability of small structural imperfections. The spatial localization of defects is exploited to identify multiple, localized, isolated defects' locations. The defects' severities are estimated based on the deviation from the circular structure's analytical mode shapes. Fast and enhanced precision of defect identification is obtained by employing the modal filtered Autoresonance technique. To validate the presented method, an experimental system consisting of a ring of coupled Helmholtz acoustic resonators was developed. Experimental results show good agreement with numerical simulations, verifying the method's capabilities to identify the location and severity of multiple defects. Thus, implementation of the suggested method provides fast and precise structural health monitoring of cyclic symmetric systems.