Elliptical Locus Research Articles

Bayesian approach of elliptical loci and RAPID for damage localization in wind turbine blade

This investigation addresses the issue of damage detection and localization in wind turbine blade laminates. This paper proposes a novel approach that integrates the elliptical trajectory and probabilistic imaging method using the Bayesian framework. This method employs multiple damage-sensitive features to enhance the reliability and robustness of sensor arrays. The algorithm is optimized by analyzing the propagation characteristics of Lamb waves in composite blade laminates. A numerical simulation is conducted on a 1.5 MW wind turbine blade laminate model, incorporating the scattered wave signal, wave arrival time, and correlation coefficient as damage characteristic signals. Markov Chain Monte Carlo sampling method is adopted to obtain the posterior distribution of the damage location and achieve accurate localization of blade delamination damage. The experimental results indicate that the damage localization algorithm, which utilizes the Bayesian approach, achieves an accuracy of approximately 97.04% in localizing delamination damage in blade laminates.

Correction to: d-elliptic loci in genus 2 and 3

Abstract We correct an error in Proposition 5.1 of the named paper, which was brought to the author’s attention by Aaron Pixton. This affects various other formulas in the paper, including Theorem 1.4, which are also corrected.

Engineering the spectra of photon triplets generated from micro/nanofiber

Quantum light sources are the core resources for photonics-based quantum information processing. We investigate the spectral engineering of photon triplets generated by third-order spontaneous parametric down-conversion in micro/nanofiber. The phase mismatching at one–third pump frequency gives rise to non-degenerate photon triplets, the joint spectral intensity of which has an elliptical locus with a fixed eccentricity of 6/3 . Therefore, we propose a frequency-division scheme to separate non-degenerate photon triplets into three channels with high heralding efficiency for the first time. Choosing an appropriate pump wavelength can compensate for the fabrication errors of micro/nanofiber and also generate narrowband, non-degenerate photon triplet sources with a high signal-to-noise ratio. Furthermore, the long-period micro/nanofiber grating introduces a new controllable degree of freedom to tailor phase matching, resulting from the periodic oscillation of dispersion. In this scheme, the wavelength of photon triplets can be flexibly tuned using quasi-phase matching. We study the generation of photon triplets from this novel perspective of spectrum engineering, and we believe that this work will accelerate the practical implementation of photon triplets in quantum information processing.

Characteristics of Elliptical Vibration-Assisted Cutting with Variations in Tilt Angle of Elliptical Locus.

Elliptical vibration-assisted cutting (EVAC), one of the advanced micromachining methods, enables results not possible with traditional ultra-precision machining. It is considered to be one of the most viable options for manufacturing micro/nanostructured surfaces. However, it is difficult to control the elliptical locus with different tilt angles; therefore, previous studies have primarily focused on fixed locus and investigated the effects of the amplitude and frequency on machining performance. In addition, tilt angle is an important factor affecting the characteristics of EVAC. To maximize the cutting performance of EVAC, the cutting characteristics of EVAC with variations in tilt angle of elliptical locus are investigated. The mathematical model of elliptical trajectory based on different tilt angles is established via geometric analysis. The effects of the different tilt angle (0-180°) on cutting forces, chip formation, defect generation and surface roughness are observed and theoretically analyzed in microgroove experiments. The experimental results show that the tilt angle has a significant effect on the cutting force, chip formation, defects and surface roughness. The best cutting performance can be obtained at the tilt angle of 30°, while the worst is recorded at 90°. The results can provide a valuable reference for further comprehensive studies to maximize the cutting performance of EVAC.

Longitudinal-torsional complex-mode ultrasonic actuator for vibration-assisted drilling of CFRP

Carbon-fiber-reinforced plastic (CFRP) composites are intensively used in aircraft and aerospace industry thanks to their superior properties. Comparing to the conventional drilling (CD), vibration-assisted drilling (VAD) is a novel machining technique suitable for drilling CFRP. Still, multi-mode excitations with elliptical locus and low vibration performance limit the applications of current VAD schemes for CFRP. To overcome these limitations and improve the overall performance, an innovative longitudinal-torsional complex-mode ultrasonic vibration-assisted actuator with single excitation and an elliptical locus is presented employing a piezoelectric transducer and a stepped horn with spiral grooves. The proposed actuator is specially designed to deliver elliptical vibration and assure high vibration performance of a tool tip. Analysis of the actuation mechanism for the longitudinal-torsional composite vibration mode is discussed, and its simplified model is developed. A detailed design process of this actuator is given. Its vibration characteristics are verified with both finite-element simulation and experimental modal analysis using a swept sine test. It is demonstrated the developed prototype achieved longitudinal-torsional elliptical vibration. To validate the machining performance of the actuator, two groups of drilling experiments were performed. These indicate that the proposed actuator is capable of drilling CFRP with improved machining performance.

Ductile machining of optical micro-structures on single crystal germanium by elliptical vibration assisted sculpturing.

Microstructure surfaces are widely and deeply applied in optical fields to decrease the volume of an optical system and realize sophisticated optical requirements. This paper studies the machining of optical micro-structures with micro-scale height on single crystal germanium by elliptical vibration assisted sculpturing (EVAS) method in which two vibration amplitudes keep constant. Due to the superimposition of low-frequency sculpture path and high-frequency elliptical locus, the investigations are focused on cutting plane to describe motion trajectory of the tool tip of cutting tool for desired complex motion path and to calculate the sculpture path of elliptical vibration cutting device to make corresponding path of tool tip be the envelope of target profile. A comparative experiment is firstly conducted to verify that elliptical locus would deteriorate machining accuracy without accurate compensation for the ellipse. Following that, 2D and 3D sinusoidal surfaces and a picture, whose structural heights are all not less than 2.0µm, are experimentally machined on single crystal germanium. Satisfying experimental results demonstrate the feasibility of EVAS method for fabricating optical micro-structures on germanium.

Elliptic vector loci of average electron velocity of electron swarm in constant-collision-frequency model gas under ac electric and dc magnetic fields crossed at arbitrary angles

The analytic solution of the average electron velocity vector $$\mathbf {W}(t)$$ of an electron swarm in gas under ac electric and dc magnetic fields ( $$\mathbf {E}(t)$$ and $$\mathbf {B}$$ , respectively) in a constant-collision-frequency model is extended to cover not only the known case of a right crossing angle ( $$\mathbf {E}(t) \perp \mathbf {B}$$ ) but also the cases of arbitrary crossing angles ( $$\mathbf {E}(t) \not \perp \mathbf {B}$$ ). The x, y, and z components of $$\mathbf {W}(t)$$ are obtained as explicit formulae with the following parameters: the amplitude E of $$\mathbf {E}(t)$$ , the angle $$\theta $$ between $$\mathbf {E}(t)$$ and $$\mathbf {B}$$ , the angular frequency $$\omega _E$$ of $$\mathbf {E}(t)$$ , the cyclotron angular frequency $$\omega _B$$ (subject to the strength B of $$\mathbf {B}$$ ), and the electron collision frequency $$\nu $$ . A basic feature that $$\mathbf {W}(t)$$ draws elliptic locus in velocity space is unchanged even under $$\mathbf {E}(t) \not \perp \mathbf {B}$$ , but the plane including the locus may tilt when $$\omega _E$$ , $$\omega _B$$ , or $$\nu $$ varies. The derivation of $$\mathbf {W}(t)$$ based on the analytic solution of single electron motion is detailed and fundamental behavior of the $$\mathbf {W}(t)$$ loci is observed to understand the electron transport under $$\mathbf {E}(t) \times \mathbf {B}$$ fields.

Invariant Center Power and Elliptic Loci of Poncelet Triangles

We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to either circumcircle or Euler’s circle is invariant, and (ii) if a triangle center of a 3-periodic in a generic nested pair is a fixed linear combination of barycenter and circumcenter, its locus over the family is an ellipse.

Localization and Imaging of Micro-Cracks Using Nonlinear Lamb Waves with Imperfect Group-Velocity Matching

Nonlinear Lamb waves have attracted increasing attention for detecting and identifying microstructural changes in structural health monitoring. However, most identification methods that determine the damage locations based on the intersections of the elliptical loci will inevitably cause positioning errors due to the change of the group velocity before and after interaction with the damage. In this work, a method focusing on elliptical rings was proposed for localization and imaging of micro-cracks in a three-dimensional structure using nonlinear Lamb waves with imperfect group-velocity matching. The width of the elliptical rings can be determined by the degree of the group-velocity mismatching of nonlinear S0 modes. The mode pair S0-s0, satisfying approximate group-velocity matching, is mainly introduced by interacting with the micro-crack. The effectiveness of the proposed methodology for damage localization is verified by the experimental testing and numerical simulation. Although the length of the being-tested small crack (about 1 mm) is smaller than the wavelength of the incident fundamental Lamb wave (around 20 mm), it can be well identified and localized using nonlinear Lamb waves. The experimental results show that the proposed method enables more reliable localization of the small crack with the crossover areas, as compared with the intersections based on the ellipse method. Furthermore, a breathing crack not situated in the propagation path can also be well localized by the proposed method in comparison with those by the probability-based diagnostic imaging in the simulation cases.

Intriguing invariants of centers of ellipse-inscribed triangles

We describe invariants of centers of ellipse-inscribed triangle families with two vertices fixed to the ellipse boundary and a third one which sweeps it. We prove that: (i) if a triangle center is a fixed affine combination of barycenter and orthocenter, its locus is an ellipse; (ii) and that over the family of said affine combinations, the centers of said loci sweep a line; (iii) over the family of parallel fixed vertices, said loci rigidly translate along a second line. Additionally, we study invariants of the envelope of elliptic loci over combinations of two fixed vertices on the ellipse.

Elliptic Loci of SU(3) Vacua

The space of vacua of many four-dimensional, $${\mathcal {N}}=2$$ supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group SU(2), this gives a fully explicit description of the low-energy effective theory in terms of an elliptic curve and associated modular fundamental domain. The two-dimensional space of vacua for gauge group SU(3) parametrizes an intricate family of genus two curves. We analyse this family using the so-called Rosenhain form for these curves. We demonstrate that two natural one-dimensional subloci of the space of SU(3) vacua, $${\mathcal {E}}_u$$ and $${\mathcal {E}}_v$$ , each parametrize a family of elliptic curves. For these elliptic loci, we describe the order parameters and fundamental domains explicitly. The locus $${\mathcal {E}}_u$$ contains the points where mutually local dyons become massless and is a fundamental domain for a classical congruence subgroup. Moreover, the locus $${\mathcal {E}}_v$$ contains the superconformal Argyres–Douglas points and is a fundamental domain for a Fricke group.

D-elliptic Loci in Genus 2 and 3

Abstract We consider the loci of curves of genus 2 and 3 admitting a $d$-to-1 map to a genus 1 curve. After compactifying these loci via admissible covers, we obtain formulas for their Chow classes, recovering results of Faber–Pagani and van Zelm when $d=2$. The answers exhibit quasimodularity properties similar to those in the Gromov–Witten theory of a fixed genus 1 curve; we conjecture that the quasimodularity persists in higher genus and indicate a number of possible variants.

A Bayesian fusion method for composite damage identification using Lamb wave

This study presents a novel method for composite damage identification using Lamb wave. A probabilistic integration of the elliptical loci method and the RAPID (reconstruction algorithm for probabilistic inspection of defects) in a Bayesian framework is proposed. The proposed method allows for the incorporation of multiple damage sensitive features in a rational manner to improve the reliability and robustness for a given array of sensors. Numerical studies are performed to verify the effectiveness of the proposed method and to compare its accuracy with existing methods. Experimental investigation using a realistic composite plate is made to further validate the proposed method. The influence of damage location and the number of participating sensors on the performance of the proposed method is discussed. Results indicate that the proposed method yields more accurate and reliable results comparing with existing methods.

On rotating vectors, Jacobi elliptic functions and free vibration of the Duffing oscillator

The free vibration displacement of an undamped hardening Duffing oscillator is described in exact form by a Jacobi elliptic function. Unlike an undamped linear oscillator, whose displacement is described by a trigonometric function, a Jacobi elliptic function is difficult to interpret by a simple inspection of the function arguments. The displacement of a linear oscillator is often visualised as a rotating vector, which has two characteristics — a constant amplitude and a phase (or frequency). These parameters are readily related to the physical response of the system. In this paper, a similar approach is applied to the free vibration displacement of a Duffing oscillator. However, the rotating vector description of the motion is much more complicated than for a linear system. It still has two characteristics though — an amplitude and a phase, but in general both these quantities are dependent on the position of the vector, i.e., they are frequency modulated. It is shown that there is not a unique rotating vector representation of the cn Jacobi elliptic function. Indeed, there are an infinite number of elliptical loci bounded between an elliptical and a circular locus of the vector. There are two specific cases. One is where the amplitude of the vector is constant and the phase angle is frequency modulated (the circle), and the other is when the amplitude of the vector is frequency modulated and the angular velocity is constant. In all other cases, both the amplitude and the angular velocity of the rotating vector are frequency modulated. To aid in the visualisation of the rotating vectors that represent the free vibration solution of a highly nonlinear hardening Duffing oscillator, two animations are provided.

How to Determine the Size of Longitudinal-flexural Mode Linear Ultrasonic Motor

Linear ultrasonic motor is more and more widely used. Based on the coupling of longitudinal and flexural mode, the piezoelectric stator induces the elliptical locus of contact points to drive the rotor. Now, how to determine the size of a linear ultrasonic motor is still an important problem. Here, we use Shi’s structure of the linear ultrasonic motor and reconstruct the finite element model by the parametric design language of APDL to execute modal, harmonic, transient and contact analyses. We find the frequency of longitudinal mode is only related to the length of the piezoelectric stator and the frequency of flexural mode is related to the length and width of the piezoelectric stator. Based on the results, we put forward a simple strategy to design the linear ultrasonic motor. We find that viscous damping is important to the coupling. Suitable viscous damping gives a large frequency band where the longitudinal and flexural modes are inspired simultaneously. Our results also give a minimum limit of the number of substeps in a period in transient analysis.

On the elliptic locus of a family of projectiles

A little-known property of free-fall motion is the elliptic locus of the maximum heights attained by coplanar projectiles launched from a single point in different directions with the same initial speed. Another, less-known, and perhaps somewhat surprising, property of this family consists in the said ellipse being also the geometric locus of the critical points of the projectiles’ distances from the launch point. In the article, to gain a better perspective on the geometry involved, we consider these loci from the standpoint of a free-falling frame. It is shown that, in this reference frame, the considered loci are congruent circles tangent to each other along the horizontal and internally tangent to another, twice as big a circle, which represents the locus of the projectiles’ ranges. This simplified geometric description is employed to recover the elliptic locus in the laboratory frame and further explore its properties.

Derivation of the electron drift velocity vector in gas under crossed ac electric and dc magnetic fields assuming constant-collision-frequency models

The periodic variation of the average electron velocity vector under crossed ac electric and dc magnetic fields is formulated analytically on the basis of kinetic equations for single-electron motion assuming constant-collision-frequency models with and without ionization. The velocity vector draws elliptic loci with the alternation of the electric field, and this is verified by Monte Carlo simulations. The amplitude and the mean deflection angle of the velocity vector, which represent basic features of electron transport in magnetized plasmas, are explicitly described with the collision frequency, the ac angular frequency, and the intensities of the fields.

Characteristic measurement of a surface acoustic wave nano-stepping motor by using a fiber-optic Michelson interferometer

The actuation technique of a surface acoustic wave motor with nanometer scale linear motion was experimentally investigated in this study. The surface acoustic wave motor comprised a stator made of a Y+128° cut, X-propagation lithium niobate substrate with silicon sliders and an array of pillar projections manufactured using semiconductor fabrication techniques. Two sets of interdigital transducers deposited on the substrate were used to generate Rayleigh waves with a driving frequency of up to 9.7 MHz. The surface acoustic wave motor was driven by friction exerted on the contact area between the slider and the surface acoustic waves in a retrogressive elliptical locus. The stepping motion of the surface acoustic wave motor was measured directly using a fiber-optic Michelson interferometer through demodulation with a digital signal processing method. A displacement of several nanometers was achieved at each step during the experiment.

Development of a Three-Degree-of-Freedom Ultrasonic Vibration Tool Holder for Milling and Drilling

This paper presents a three-degree-of-freedom (3-DOF) ultrasonic vibration tool holder for milling and drilling operations. The 3-DOF vibrations are generated by a novel actuator that contains three groups of piezoelectric rings actuating in X -, Y- , and Z -directions at the natural frequencies of the structure. The vibrations in XY plane are excited at the frequency but with slightly different amplitudes to produce an elliptical locus for intermittent tool-workpiece contact, which assists milling process, and the vibrations along Z -axis are used in drilling operations. The 3-DOF ultrasonic vibration actuator excites the structure's natural frequencies, which are above 16 kHz with amplitudes up to 20–25 $\mu$ m. A control system has been developed for tracking the resonance frequencies during cutting from the phase of actuator's impedance. Preliminary cutting tests show that the cutting forces are reduced by over 30% with the ultrasonic vibration assisted milling and drilling operations.

Experimental Study of Microgroove Surface Using Three-Dimensional Elliptical Vibration Texturing

This paper presents our preliminary study of the microstructured surface on microgrooves patterns, which have manufactured using the three-dimensional elliptical vibration texturing (3D-EVT) method. The 3D-EVT method uses a three-dimensional trajectory of elliptical locus of the cutting tool tip in high vibration frequency to fabricate the microstructure or the vibration mark pattern on the machined surface. The experimental study has been done with a variation of the nominal cutting speeds from 300 to 600 mm/min. The preliminary results showed that in the low nominal cutting speed of 300 and 450 mm/min, the microstructured surfaces were not manufactured well due to the bulging existence on the microstructured surface in which it is correlated with the build-up edge phenomena. On the other hand, a relative excellent microstructured surface can be achieved at the higher nominal cutting speed of 600 mm/min. In addition, a lubricant could be used to obtain an excellent microstructured surface to avoid the built-up edge phenomena.