Resonance Equation Research Articles

Design of a damped vibration absorber to control the resonant vibration of roll

Strip rolling is currently the main steel production process. The vertical vibration of the rolling mill caused by the large reduction ratio and high rolling speed reduces the qualification rate of strip products, and even damages the rolling equipment. In this paper, a new particle damping absorber (PDA) is designed to suppress the vertical vibration of the mill. The adaptive genetic algorithm (AGA) is used to optimize the basic parameters of PDA. Considering the nonlinear stiffness between the roll and the rolling stock, a four-degree-of-freedom vertical vibration model of the roll system with the PDA is established. The amplitude-frequency characteristic equations of the primary resonance and the internal resonance are solved by using the multi-scale method. The static bifurcation of the primary resonance of the system is investigated based on the singularity theory. The effects of parameters on vibration are analyzed. The control effect of the vibration absorber is investigated experimentally, and it is found that the experimental results are basically consistent with the theoretical analysis, which verifies the reliability of the established vibration model. The results show that the PDA can effectively suppress the vertical vibration of the roll. It provides some theoretical guidance for the research on vibration suppression of rolling mills.

Unified Bidirectional Resonant Frequency Tracking for CLLC Converters

Unregulated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLLC</i> converters have significant potential for dc transformers and two-stage battery chargers due to their high efficiency and bidirectional capability. In these applications, unregulated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLLC</i> converters are usually designed to have their switching frequency fixed at resonance for maximum efficiency. However, the resonant frequency drifts from the designed value due to resonant tank parameter deviations, which leads to a decrease in efficiency for unregulated converters. To address this issue, this article proposes the bidirectional resonant frequency and a unified tracking algorithm. By analyzing the generalized resonant frequency equations in depth, an interesting feature is discovered for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLLC</i> converters: forward mode and backward mode achieve maximum efficiency at an identical (but not constant) frequency. Thus, this unique frequency is defined as bidirectional resonant frequency (BRF). As the BRF varies with parameter deviations, a unified bidirectional tracking algorithm is proposed, which unifies the tracking of bidirectional maximum efficiency points into the tracking of a BRF. The introduction of the BRF combined with the unified tracking algorithm provides a simple, accurate, and low-cost solution for bidirectional resonance tracking of unregulated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLLC</i> converters. Finally, experimental results show that unified bidirectional tracking and efficiency improvement are achieved under parameter deviations.

Stability of the Dynamical Motion of a Damped 3DOF Auto-parametric Pendulum System

PurposeThe motion of three degrees-of-freedom (DOF) of an automatic parametric pendulum attached with a damped system has been investigated. The kinematics equations of this system have been derived employing Lagrange’s equations in accordance to it’s the generalized coordinates.MethodsThe method of multiple scales (MMS) has been used to obtain the solutions of the controlling equations up to the third-order of approximation. The solvability criteria and modulation equations for primary external resonance have been explored simultaneously.ResultsThe non-linear stability approach has been used to analyze the stability of the considered system according to its different parameters. Time histories of the amplitudes and the phases of this system have been graphed to characterize the motion of the system at any given occurrence.ConclusionsThe different zones of stability and instability of this study have been checked and examined, in which the system's behavior has been revealed to be stable for various values of its variables.

High order rational solitons and their dynamics of the 3-wave resonant interaction equation

In this work, we derive a family of high order rational solitons of the 3-wave resonant interaction (3WRI) equation by utilizing generalized Darboux-dressing transformation. We observe that breathers in the three components display self-similar behaviors in the process of propagation. For the first order rational soliton, it is interesting that the dark soliton component bifurcates so as to generate a peak whose amplitude is three times higher than the surrounding background itself with an infinite life time. It would provide us a way to generate ‘eternal’ large amplitude wave in weakly nonlinear dispersive media. Furthermore, the interactions of two rational solitons are studied.

Semi-analytic semi-numerical analysis of dynamic characteristics of a two-stage coupled series PGT based on IHB and MsP methods

This paper conducts an analytical investigation of the dynamic response characteristics of a two-stage coupled series system (TsCSS) with a planetary gear transmission (PGT), consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method, in order to verify the feasibility and efficiency of the improved method. Afterwards, the multi-scale perturbation (MsP) method is applied to the proposed TsCSS. The frequency response equations of the primary resonance, subharmonic resonance and superharmonic resonance, are solved in order to generate the frequency response characteristic curves of the planetary transmission system. A comparison between the results obtained by the MsP method and the numerical integration method demonstrates that the former is ideal and credible in most of the regions. Considering the parameters of TsCSS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The impact of the time-varying parameters and nonlinear deenthing caused by the gear teeth clearance on the amplitude–frequency characteristics of the TsCSS components, is finally studied.

A study of resonance Y-type multi-soliton solutions and soliton molecules for new (2+1)-dimensional nonlinear wave equations

&lt;abstract&gt; &lt;p&gt;In this study, a fourth-order nonlinear wave equation with variable coefficients was investigated. Through appropriate choice of the free parameters and using the simplified linear superposition principle (LSP) and velocity resonance (VR), the examined equation can be considered as Hirota–Satsuma–Ito, Calogero–Bogoyavlenskii–Schiff and Jimbo–Miwa equations. The main objective of this study was to obtain novel resonant multi-soliton solutions and investigate inelastic interactions of traveling waves for the above-mentioned equation. Novel resonant multi-soliton solutions along with their essential conditions were obtained by using simplified LSP, and the conditions guaranteed the existence of resonant solitons. Furthermore, the obtained solutions were used to investigate the dynamic and fission behavior of Y-type multi-soliton waves. For an accurate investigation of physical phenomena, appropriate free parameters were chosen to ascertain the impact on the speed of traveling waves and the initiation time of fission. Three-dimensional and contour plots of the obtained solutions are presented in &lt;xref ref-type="fig" rid="Figure1"&gt;Figures 1&lt;/xref&gt;–&lt;xref ref-type="fig" rid="Figure6"&gt;6&lt;/xref&gt;. Additionally, two nonlinear equations were formulated and investigated using VR, and the related soliton molecules were simultaneously extracted. The reported resonant Y-type multi-soliton waves and equations are new and have not been previously investigated. They can be used to explain modeled physical phenomena and can provide information about dynamic behavior of shallow water waves.&lt;/p&gt; &lt;/abstract&gt;

Optimisation of the geometry of axisymmetric point-absorber wave energy converters

We propose a scientifically rigorous framework to find realistic optimal geometries of wave energy converters (WECs). For specificity, we assume WECs to be axisymmetric point absorbers in a monochromatic unidirectional incident wave, all within the context of linearised potential theory. We consider separately the problem of a WEC moving and extracting wave energy in heave only and then the more general case of motion and extraction in combined heave, surge and pitch. We describe the axisymmetric geometries using polynomial basis functions, allowing for discontinuities in slope. Our framework involves ensuring maximum power, specifying practical motion constraints and then minimising surface area (as a proxy for cost). The framework is robust and well-posed, and the optimisation produces feasible WEC geometries. Using the proposed framework, we develop a systematic computational and theoretical approach, and we obtain results and insights for the optimal WEC geometries. The optimisation process is sped up significantly by a new theoretical result to obtain roots of the heave resonance equation. For both the heave-only, and the heave-surge-pitch combined problems, we find that geometries which protrude outward below the waterline are generally optimal. These optimal geometries have up to 73 % less surface area and 90 % less volume than the optimal cylinders which extract the same power.

Microwave excitations and hysteretic magnetization dynamics of stripe domain films

FeNi films with the stripe domain (SD) pattern are prepared by electrodeposition and sputtering methods. The magnetic domain, static magnetic parameters, and quality factor, as well as dynamic properties of the two films, are respectively performed. The results show the magnetizations of the film were dependent on the direction of SD, and the rotation of the SD is lagging behind the magnetization reversal. The microwave properties of the SD emerge dynamic hysteresis before the saturation magnetic field. These microwave properties are selectively excited with acoustic mode, optical mode, and spin-wave mode. The frequency and intensity of different resonance modes of stripe domain are determined by the local magnetization. The magnetization variations and the rotation of SD of different modes are further illuminated by the micromagnetic simulation. The magnetic anisotropy and the resonance intensity of permeability of different modes were finally described by the modified resonance equations.

On the use of nonlinear impact oscillators in vibrating electromagnetic based energy harvesters

This work investigates the use of impact-oscillators in nonlinear electro-magnetic cantilever based broad bandwidth frequency energy harvester. The electro-mechanical model includes the dynamical equations of the nonlinear impact-oscillators, the nonlinear frequency resonant cantilever beam equation of the harvester, and the Lorenz magnetic force resulting from the surrounding nonlinear magnetic field. The simulated results of the harvested power show a band of resonant frequencies of the vibrating cantilever beam extended beyond its fundamental natural frequency. Due to the strong nonlinearity of both the impact oscillators’ forces and the magnetic flux, the harvested power of the enhanced system shows wider bands in its respective frequency response. Moreover, the harvested power of the cantilever beam shows that considering two magnets in repulsion generates higher values mainly due to the strong nonlinear transient magnetic flux, resulting into few times more power as compared to the case of considering only one magnet in the system.

Thermoacoustics of multi-burner combustors with plenum and chamber cross-talk

This paper presents a thermoacoustic model in the frequency domain for multi-burner combustors composed of multiple plenum-nozzle-chamber modules with cross-talk at both plenums and chambers. The thermoacoustic model is given as a multi-input-multi-output transfer function matrix from heat fluctuation input vector to velocity fluctuation output vector. The discrete rotational symmetry of a multi-burner combustor is exploited to obtain a simple, diagonal transfer function matrix. The resonance equation, whose roots are resonances, is given as a set of scalar equations. The explicit form of the resonance equation reveals how combustor and operating parameters influence the resonances, and which parameters are significant for the longitudinal-azimuthal coupling and the plenum-chamber coupling. The diagonalization also allows us to show constructively that every mode shape is a Bloch wave and those mode shapes can characterize all resonances in terms of longitudinal and azimuthal mode numbers. At a low frequency, the resonance equation can be approximated as biquadratic and quadratic equations for a typical combustor geometry and operating conditions. A bunch of low-frequency resonances, peculiar to multi-burner combustors with cross-talk, are interpreted as azimuthal Helmholtz modes in which plenums/chambers and cross-talk ducts serve as the cavities and necks, respectively. Our combustor model is combined with a diagonal flame transfer function matrix for combustion instability analysis and we develop a set of scalar stability equations whose roots determine combustion instability. This stability equation reveals how combustion instability depends on the combustor geometry and operating parameters. Several numerical examples are presented to validate our thermoacoustic model and theoretical findings.

Graphene\u2019s non-equilibrium fermions reveal Doppler-shifted magnetophonon resonances accompanied by Mach supersonic and Landau velocity effects

Oscillatory magnetoresistance measurements on graphene have revealed a wealth of novel physics. These phenomena are typically studied at low currents. At high currents, electrons are driven far from equilibrium with the atomic lattice vibrations so that their kinetic energy can exceed the thermal energy of the phonons. Here, we report three non-equilibrium phenomena in monolayer graphene at high currents: (i) a “Doppler-like” shift and splitting of the frequencies of the transverse acoustic (TA) phonons emitted when the electrons undergo inter-Landau level (LL) transitions; (ii) an intra-LL Mach effect with the emission of TA phonons when the electrons approach supersonic speed, and (iii) the onset of elastic inter-LL transitions at a critical carrier drift velocity, analogous to the superfluid Landau velocity. All three quantum phenomena can be unified in a single resonance equation. They offer avenues for research on out-of-equilibrium phenomena in other two-dimensional fermion systems.

Sign-Changing Solutions for Resonant and Superlinear Nonhomogeneous Elliptic Equations

Sign-Changing Solutions for Resonant and Superlinear Nonhomogeneous Elliptic Equations

Construction of new exact solutions of the resonant fractional NLS equation with the extended Fan sub-equation method

The fucus of this study is to find a set of some novel solutions concerning the resonant fractional nonlinear Schrödinger equation (R-FNLSE) with quadratic-cubic nonlinearity by employing the extended FAN sub-equation approach. The parameter α∈(0,1] is the core constraint which simulate the flow rate propagation, plays a key rote in telecommunications and the theory of optical fibres. This equation expresses the gesture of solitons and Madelung fluids in various nonlinear systems. These outcomes are optical, bright, dark, explicit, periodic and combined wave solutions and efficiently demonstrated with the aid of 3D plots. The representation of these solutions is carried out in order to have an idea about the structure of such model and can also be extended to many other complex models of recent era.

Research on vibration characteristics of resonator for the HRG

Resonator is the core structure of hemispherical resonator gyro (HRG), and its vibration characteristics play an important role in the output accuracy of gyro. A method combining dynamics model and finite element method (FEM) is proposed to study vibration characteristics of resonator. Firstly, this paper established dynamics equations of ideal resonator and dynamics equations of resonator with density distributed nonuniformity, and gave the mathematical expression of frequency split. Then, accomplished modal analysis and obtained natural frequency, mode shapes of resonator based on ANSYS Workbench. Finally, established the resonator model in consideration of density distributed nonuniformity by FEM, and investigated the influnce of the 4th harmonic of density nonuniformity on resonator’s frequency split. The finite element analysis was in good agreement with the results calculated by dynamics equations, which verified the accuracy of dynamics model.

Semi-numerical analysis of a two-stage series composite planetary transmission considering incremental harmonic balance and multi-scale perturbation methods

Abstract. This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, which solves the closed solution of incremental parameters passing through the singularity point of the analytical path, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this subject to analyze the parameter excitation and gap nonlinear equations and then to obtain the analytical frequency response functions including the fundamental, subharmonic, and superharmonic resonance responses. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary gear system (PGS) in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude–frequency characteristics of TsSCS components are studied in this special topic.

Morse inequalities at infinity for a resonant mean field equation

In this paper, we study the following mean field type equation: [Formula: see text] where [Formula: see text] is a closed oriented surface of unit volume [Formula: see text] = 1, [Formula: see text] positive smooth function and [Formula: see text], [Formula: see text]. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A “critical point at infinity” approach, Discrete Contin. Dyn. Syst. 37(4) (2017) 1789–1818] we develop, under generic condition on the function K and the metric g, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters [Formula: see text]. We derive from these Morse inequalities at infinity various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. [Formula: see text].

Semi-numerical analysis of a two-stage series composite planetary transmission considering IHB and MsP methods

This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this paper. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary transmission system in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude-frequency characteristics of TsSCS components are studied in this special topic.

Nonlinear Vibration Analysis of a Cantilever Beam with a Transverse Fatigue Crack

Nonlinear phenomena widely exist in engineering applications. A common example of this phenomenon in aerospace structures is the creation of fatigue cracks that open and close continuously under dynamic loads during the vibration cycle, causing a bilinear behavior in the structure. Late detection of such cracks can lead to a catastrophic failure. Therefore, identifying the nonlinear behavior of the cracked structure is very important for the prevention of structural failures. In this study, the nonlinear vibration behavior of a cantilever beam with a transverse breathing crack with bilinear behavior has been studied. For this purpose, we first estimate the polynomial function equivalent to the bilinear function of the beam stiffness. Then, using the well-known perturbation method, the method of multiple scales, the nonlinear equation of the beam is solved, and the frequency response equations for both harmonic and superharmonic resonances are extracted. Then, the sensitivity of the responses to the crack parameters such as depth, location, excitation force, and damping coefficient are investigated. The beam frequency response curves in the main resonance have become highly nonlinear due to the increase of the crack parameters and causes softening of the curves. Also, according to the results of the beam vibrations in superharmonic resonance, it was observed that the behavior of the beam in superharmonic resonance has a higher sensitivity to the existence of the breathing crack.

A planar microwave resonator with odd resonance for calibration in permanent moisture sensing applications

A cylindrical dual-mode planar microwave ring resonator for a permanent moisture sensor is presented. The ring is design for TMn10 fundamental (odd-mode) resonance, and optimized to sustain stability (suitable for calibration reference against environmental interference) using Ansys high-frequency structure simulator. An even-mode resonance is perturbed by etching a step-impedance stub on the upper 0.5λ ring. The equations for the TMn10 mode resonance and its mode splitting are stated. The equivalent circuits of the ring, odd-, and even-modes are derived. The even-mode is set to capture the response of the permanent moisture withholding hydrophilic polyvinyl alcohol (PVA)–polyaniline (PANI) 0.26 mm-thick films. The in-lab made films are produced by infusing certain concentrations of PANI into pure PVA. On-spot measurements using a vector network analyzer (VNA) and time-based continuous data acquisitions using LabVIEW on NI PXIe–1075 VNA are done to confirm repeatability of the obtained results. A measured permanent moisture response of 180 MHz is confirmed for 50% PANI with a thickness (hfilm) of 0.55 mm. The sources of permanent moisture withholding capability of the films are investigated. Structural observation of the residue films, its permittivity changes, and morphological structural imagery confirmed the sources of permanent response.

Soliton gas in bidirectional dispersive hydrodynamics.

The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and overtaking collisions of solitons take place. We distinguish between two qualitatively different types of bidirectional soliton gases: isotropic gases, in which the position shifts accompanying the head-on and overtaking soliton collisions have the same sign, and anisotropic gases, in which the position shifts for head-on and overtaking collisions have opposite signs. We construct kinetic equations for both types of bidirectional soliton gases and solve the respective shock-tube problems for the collision of two "monochromatic" soliton beams consisting of solitons of approximately the same amplitude and velocity. The corresponding weak solutions of the kinetic equations consisting of differing uniform states separated by contact discontinuities for the mean flow are constructed. Concrete examples of bidirectional Eulerian soliton gases for the defocusing nonlinear Schrödinger (NLS) equation and the resonant NLS equation are considered. The kinetic equation of the resonant NLS soliton gas is shown to be equivalent to that of the shallow-water bidirectional soliton gas described by the Kaup-Boussinesq equations. The analytical results for shock-tube Riemann problems for bidirectional soliton gases are shown to be in excellent agreement with direct numerical simulations.