Resonance Equation Research Articles

Propagation of Optical Solitons to the Fractional Resonant Davey-Stewartson Equations

Propagation of Optical Solitons to the Fractional Resonant Davey-Stewartson Equations

A novel theoretical analysis method for the longitudinal cascaded piezoelectric transducer based on equivalent circuit

Abstract High power ultrasonic transducers are widely used in the fields of ultrasonic cleaning, ultrasonic welding and therapeutic ultrasound. To further improve the ultrasonic intensity and power of piezoelectric transducers for power ultrasound applications, the cascaded piezoelectric transducer have received tons of attention. The electromechanical equivalent circuit, as a classical transducer analysis method, translates the structural and mechanical parameters into the corresponding circuit parameters and helps us to better understand and analyze the performance characteristics of the transducer, which is of great significance for the design and optimization of the transducer. In this paper, a new theoretical analysis method based on the equivalent circuit is proposed for cascaded piezoelectric transducer. In this method, the equivalent circuit of each component of the longitudinal piezoelectric transducer is cascaded, and equivalent circuit is analyzed using the Kirchhoff’s law to obtain the frequency resonance equation of the piezoelectric transducer. This analysis method avoids the complex equivalent transformation of the equivalent circuit including multiple excitation sources and also obtains the vibration velocity of the piezoelectric transducer radiation surfaces, more information on the vibration characteristics of the transducer was gained from the theoretical aspect. Finally, the theoretical analysis results are compared with the finite element simulation analysis and the traditional analysis results for verification.

Bifurcation study, phase portraits and optical solitons of dual-mode resonant nonlinear Schrodinger dynamical equation with Kerr law non-linearity

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

Existence of Solution for a Katugampola Fractional Differential Equation Using Coincidence Degree Theory

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance -(Da+α,ρx)(t)=f(t,x(t),Da+α-1,ρx(t)),t∈(a,b),x(a)=0,x(b)=∫abx(t)dA(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -(D^{\\alpha ,\\rho }_{a+}x)(t)= & {} f(t,x(t),D^{\\alpha -1, \\rho }_{a+}x(t)), \\ \\ t\\in (a,b), \\\\ x(a)= & {} 0, \\ \\ x(b)=\\int _{a}^{b} x(t){\ ext {d}}A(t), \\end{aligned}$$\\end{document}where 1<α≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<\\alpha \\le 2$$\\end{document}, and Da+α,ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{\\alpha ,\\rho }_{a+}$$\\end{document} is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and ∫abx(t)dA(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{a}^{b} x(t){\ ext {d}}A(t)$$\\end{document} denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results.

Variational theory for the resonant T-curvature equation

Variational theory for the resonant T-curvature equation

Nonlinear combination resonance analysis of parametric-forced excitation for an axially moving piezoelectric rectangular thin plate in thermal- electromechanical coupling field

In this paper, the combination resonance of parametric-forced excitation characteristics for an axially moving rectangular piezoelectric plate under a thermal-electromechanical field is studied. Based on Kirchhoff-Love plate theory and Von Karman theory, the transverse vibration governing equations are derived from Hamilton's principle. The equations are discretized to ordinary differential equations by the Galerkin method. Then, the multiple scales method is applied to solve the system combination resonance equation, two different resonance states and corresponding amplitude-frequency response equations are obtained by eliminating the secular term, respectively. Additionally, the stability of the steady-state responses is analyzed by Lyapunov stability. Based on the numerical analysis, the influence of axial velocity, external voltage, central temperature difference, structural damping, and other parameters on nonlinear combination resonance response are investigated. The effect of parameter variation on period-doubling bifurcation and the chaotic motion of the system are also discussed by the system global bifurcation diagram.

Combined resonant tunneling and rate equation modeling of terahertz quantum cascade lasers

Terahertz (THz) quantum cascade lasers (QCLs) are technologically important laser sources for the THz range but are complex to model. An efficient extended rate equation model is developed here by incorporating the resonant tunneling mechanism from the density matrix formalism, which permits to simulate THz QCLs with thick carrier injection barriers within the semi-classical formalism. A self-consistent solution is obtained by iteratively solving the Schrödinger–Poisson equation with this transport model. Carrier–light coupling is also included to simulate the current behavior arising from stimulated emission. As a quasi-ab initio model, intermediate parameters, such as pure dephasing time and optical linewidth, are dynamically calculated in the convergence process, and the only fitting parameters are the interface roughness correlation length and height. Good agreement has been achieved by comparing the simulation results of various designs with experiments, and other models such as density matrix Monte Carlo and non-equilibrium Green's function method that, unlike here, require important computational resources. The accuracy, compatibility, and computational efficiency of our model enable many application scenarios, such as design optimization and quantitative insights into THz QCLs. Finally, the source code of the model is also provided in the supplementary material of this article for readers to repeat the results presented here, investigate, and optimize new designs.

Low-Temperature Airflow-Drying Cracking Coupling with Flexible Mechanical Deshelling for Camellia oleifera Fruit: Process and Mechanism

Addressing the problems of high seed-breaking ratio and difficult seed-shell separation in the current mechanical deshelling of Camellia Oleifera fruit (COF), this work innovatively proposed a strategy of low-temperature airflow-drying cracking coupling with flexible mechanical deshelling. The cracking ratio of COF reached 100% after airflow-drying at 20–50 °C for 4–10 h. Moreover, the tissue structure of the Camellia shell remained unchanged after airflow drying. The mechanical shelling ratio of cracked fruits (cracking gap 6 mm) reached 98.37%, and the shells present as integrated lamelliform for facilitating later sorting. The results of low-field nuclear magnetic resonance (LF-NMR) and shrinkage equation showed that the free water of outer layers in the Camellia shell evaporated first, then the free water of inner layers transferred to outer layers to further evaporate, creating a moisture gradient and moisture stress to cause the shrinkage of shell and cracking of COF.

Active Acoustic Metamaterial Based on Helmholtz Resonators to Absorb Broadband Low-Frequency Noise.

The aim of the present work is to design active acoustic metamaterial consisting of an array of Helmholtz resonators and fabricating them using an additive manufacturing technique in order to assist in a reduction in noise levels in aerospace applications. To this aim, initially, a passive metamaterial consisting of an array of 64 Helmholtz resonator unit cells is designed and tested to establish the effectiveness and region of performance. The selected design variable for change is identified as the resonator cavity depth through the frequency response for each parameter of the Helmholtz resonance equation and randomized to achieve a broadband frequency range of the passive metamaterial. An active model of this design (actuated by a stepper motor) is fabricated and tested. The metamaterials are tested under two acoustic set-ups: a closed system aimed at recreating the environment of a soundproof room and an open-system aimed to recreate the condition of an active liner. For the case of passive system, the metamaterial gave sound attenuation of 18 dB (for f = 150 Hz) in open system configuration and 33 dB (f = 350 Hz) in closed system configuration. The attenuation obtained for the active model was 10-15 dB over the mean line performance for the case of closed system and 15-20 dB for the case of open system. The closed system was also tested for performance at multiple cavity depths by setting two wall depths at 10 mm and three walls at 50 mm. This test yielded an attenuation of 15 dB at 180 Hz, the frequency corresponding to 50 mm cavity depth, and 10 dB at 515 Hz, corresponding to 10 mm cavity depth.

Resonance analysis and time-delay feedback controllability for a fractional horizontal nonlinear roller system

&lt;p&gt;In this paper, we investigated the nonlinear vibration characteristics and time-delay feedback controllability of a fractional horizontal roll system, which is described by a fractional Duffing-van der Pol oscillator under an external harmonic excitation. We focused on the resonance of fractional roller systems and conducted corresponding vibration control. The amplitude-frequency equations of primary resonance and superharmonic resonance were obtained using the multiple scale method. The amplitude-frequency characteristic curves of the system with different parameters were presented, and the influence of system parameters on the curves was analyzed. In addition, the time-delay feedback controller was designed to control the parameter excitation vibration. The numerical simulation results have verified the effectiveness of the time-delay controller in eliminating the jumping and hysteresis phenomena of the rolling system. The comparisons of approximate analytical solution and numerical solution was fulfilled, and the result certifies the correctness and satisfactory precision of the approximately analytical solution. The analysis results provide certain theoretical guidance for the vibration reduction of the horizontal nonlinear roller system.&lt;/p&gt;

Existence and multiplicity of periodic solutions for a nonlinear resonance equation with singularities

We investigate a second-order periodic system with singular potential and resonance. Utilizing the main integral method and fixed point theorems, we establish the existence and multiplicity of periodic solutions with respect to time under certain assumptions on the unbounded or oscillatory term.

Low-light Image Enhancement Method Based on Illumination Curve Adjustment

This paper presents an advanced low-light image enhancement approach based on Zero-Reference Deep Curve Estimation. The technique enhances images through: (1) the integration of the Dynamic Stochastic Resonance equation for enhanced contrast and brightness, leveraging adjustable parameters for optimal image quality, and (2) application of an Anisotropic Diffusion function for image denoising and smoothing, preserving edges and texture details while effectively removing noise. This adaptive algorithm tailors diffusion intensity to local image features, further refining the resulting image quality. Experimental results confirm the efficacy of our approach in improving visual quality and noise removal in low-light images.

Development of a low-frequency accelerometer based on symmetric multi-stage triangular beam fiber Bragg grating

In order to meet the needs of accelerometers for low-frequency seismic exploration, a fiber-optic grating accelerometer based on a symmetric multi-stage triangular hinge is proposed. The sensitivity of the accelerometer is improved by the symmetrical multi-stage triangular beam structure, and experimental verification is carried out. The resonant frequency and sensitivity equations of the accelerometer are given based on theoretical analysis, while the finite element analysis software is used to simulate and analyze the characteristics of the accelerometer such as resonant frequency, stress distribution, and amplitude-frequency response. The structural parameters of the accelerometer are optimized by numerical analysis software, and the influence of the structural parameters on the sensitivity and resonant frequency of the accelerometer is analyzed. The experimental results show that the accelerometer has a resonant frequency of 140 Hz and a linearity of 99.9 % in the flat region of 1–70 Hz, a sensitivity of 337.325 pm/g, and an interference immunity of 5.39 % in the lateral direction in the working axis direction. This indicates that the accelerometer shows good linearity and lateral interference immunity in the flat interval, which well meets the needs of seismic wave signal monitoring in the field of low-frequency oil and gas exploration.

On weakly turbulent solutions to the perturbed linear harmonic oscillator

abstract: We introduce specific solutions to the linear harmonic oscillator, named bubbles. They form resonant families of invariant tori of the linear dynamics, with arbitrarily large Sobolev norms. We use these modulated bubbles of energy to construct a class of potentials which are real, smooth, time dependent and uniformly decaying to zero with respect to time, such that the corresponding perturbed quantum harmonic oscillator admits solutions which exhibit a logarithmic growth of Sobolev norms. The resonance mechanism is explicit in space variables and produces highly oscillatory solutions. We then give several recipes to construct similar examples using more specific tools based on the continuous resonant (CR) equation in dimension two.

Lithography-free high sensitivity perfect absorption based on Graphene/α-MoO3/SiC and Tamm plasmonic structure

In this paper, we propose a tunable perfect-absorption lithography-free refractive index sensor structure, which couples the Tamm phonon polariton (TPhP) mode and Graphene/α-MoO3/SiC (GMS) mode. By aligning the modal frequencies of TPhP and GMS, the coupling of the two modes is realized with perfect absorption. We utilize the transfer matrix method (TMM), resonance equation and coupled mode theory (CMT) to theoretically analyze the absorption of different modes and apply the structure to refractive index sensor. By modulating the structure parameter, the sensitivity of the sensor can be as high as 6 μm/RIU. This work could contribute to several fields such as refractive index sensors, photodetectors, absorbers, and energy harvesting devices.

Statistical solutions and Kolmogorov entropy for the lattice long-wave–short-wave resonance equations in weighted space

This article studies the lattice long-wave–short-wave resonance equations in weighted spaces. The authors first prove the global well-posedness of the initial value problem and the existence of the pullback attractor for the process generated by the solution mappings in the weighted space. Then they establish that the process possesses a family of invariant Borel probability measures supported by the pullback attractor. Afterwards, they verify that this family of Borel probability measures satisfies the Liouville theorem and is a statistical solution of the lattice long-wave–short-wave resonance equations. Finally, they prove an upper bound of the Kolmogorov entropy of the statistical solution.

Quasi-periodic solutions for quintic completely resonant derivative beam equations on T2

In the present paper, we consider two dimensional completely resonant, derivative, quintic nonlinear beam equations with reversible structure. Because of this reversible system without external parameters or potentials, Birkhoff normal form reduction is necessary before applying Kolmogorov–Arnold–Moser (KAM) theorem. As application of KAM theorem, the existence of partially hyperbolic, small amplitude, quasi-periodic solutions of the reversible system is proved in this paper.

High Precision Mass Balancing Method for the Fourth Harmonic of Mass Defect of Fused Quartz Hemispherical Resonator Based on Ion Beam Etching Process

In this paper, a novel ultra-high precision mass balancing method for fused quartz hemispherical resonator based on the ion beam etching process is proposed from the perspective of vibration mechanics. Firstly, the motion equations of resonator with mass defect are established to reveal the quantitative relationship between the amplitude of defective mass and frequency split, and the geometrical relationship between the distribution of defective mass and eigenfrequency axis, which is the equivalent morphology of mass defect. Secondly, according to the motion equations, the radial vibration model of resonator lip edge is deduced, then a new identification model for the vibration parameters composed of the frequency split, mechanical time constant, and azimuth of eigenfrequency axis based on the vibration envelope and spectral analysis is established. The vibration parameters are identified by nonlinear optimization method. After that, the mass removal function and etching efficiency function of ion beam acting on the resonator are calibrated by experimental data, then the etching trajectory is planned and the etching time is calculated. Finally, the experimental result of mass balancing indicates that the frequency split is reduced from 0.2354Hz to 0.0006Hz, and the quality factor is greater than 2.2×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sup> , which can significantly improve the vibration performance of resonator.

Resonant fractional order differential equation with two-dimensional kernel on the half-line

Resonant fractional order differential equation with two-dimensional kernel on the half-line

Auto-parametric resonance of flexible viscoelastic beams under interaction between longitudinal and transverse modes

For a system encompassing subsystems, dynamic modal interaction may experience unanticipated vibrations among subsystems that are excited indirectly. This paper focuses attention to a nonlinear auto-parametric resonance phenomenon of a flexible beam introduced via modal coupling of longitudinal forced resonance and transverse parametric resonance under longitudinal pulsatile excitation. The coupled resonance equations are first modeled based on the Euler-Bernoulli beam theory by incorporating nonlinear curvature, von Kármán's nonlinear strain-displacement relationship and the flexible beam model characterizing nonlinear inertia and displacement interaction. The Galerkin truncation-incremental harmonic balance method (IHBM) is then introduced to obtain both the stability boundary and amplitude-frequency bifurcation maps. Through the corresponding relationship between stability boundary and bifurcation diagram, the energy exchange between two modes is identified as well as the instability region and bifurcation topology can be tuned. It is found that the modal interaction greatly widens the instability region with at most two resonance bands of lateral resonance. By tuning structural and foundation damping, resonance can be controlled to occur only in one frequency band of the pair as desired. This research provides guidance to design broadband energy harvesting or vibration suppression equipment with tunable working frequency band for using the strategy of auto-parametric resonance.