Non-resonant Conditions Research Articles

Examining the behavior of a two-connected-spring pendulum according to non-resonance and resonance conditions

This paper focuses on the planar forced oscillations of a particle linked to its support point through a system comprising two nonlinear springs arranged in series with two viscous dampers. The motion of this point is restricted to being on an elliptic route. Three degrees-of-freedom (DOF) characterize the system’s motion, explained by two differential equations (DEs) and another algebraic equation derived using Lagrange’s equations. The traditional method of multiple-time scales (MMTS) is applied in the context of the time realm. The dynamics of forced and damped oscillations is explored in this work, highlighting non-resonant conditions along with two distinct external resonances. Furthermore, stationary periodic states influenced by external resonances are examined, and the stability of each case is evaluated. The asymptotic solutions of the controlling equations are solved up to the third approximation. The classification of resonance cases is based on the second and third order of the approximate solutions. Therefore, the solvability criteria and the modulation equations (ME) are obtained. The Routh–Hurwitz conditions (RHCs) serve as the basis for assessing the stability and instability criteria. A discussion and graphical representations of the time histories, frequency response curves (FRC), and stability regions are provided to demonstrate the beneficial effects of different physical parameter inputs on the system’s performance. The obtained outcomes can improve the performance of spring-based mechanical systems like vibrational isolation devices, shock absorbers, and suspension mechanisms. Additionally, they have the potential to enhance stability and control in robotic limbs and aircraft landing gear systems, ensuring better stress absorption and stability.

Periodic attitude motions of an axisymmetric spacecraft in an elliptical orbit near the hyperbolic precession

This paper deals with the existence and stability of periodic attitude motion near hyperbolic precession (HP) for a dynamically symmetric rigid body (RB) with its center of mass moving along an elliptical orbit. We present the definitions of undisturbed and disturbed periodic attitude motions, and approximate analytical solutions for them are derived using the multiscale method under non-resonance, internal resonance, and combination resonance conditions. Based on the definition of Lyapunov stability, we demonstrate stability in the non-resonant case and establish the conditions require for stability in the internal resonance scenario and a numerical analysis supports the theoretical results. The findings of this study extend classical results on RB dynamics in central gravitational fields. These insights demonstrate that appropriate parameter selection can achieve stable periodic attitude motion for RBs is in elliptical orbits, offering practical implications for spacecraft attitude control and mission reliability.

Enhancing Resonant Second-Harmonic Generation in Bilayer WSe2 by Layer-Dependent Exciton-Polaron Effect.

Two-dimensional (2D) materials serve as exceptional platforms for controlled second-harmonic generation (SHG). Current approaches to SHG control often depend on nonresonant conditions or symmetry breaking via single-gate control. Here, we employ dual-gate bilayer WSe2 to demonstrate an SHG enhancement concept that leverages strong exciton resonance and a layer-dependent exciton-polaron effect. By selectively localizing injected holes within one layer, we induce exciton-polaron states in the hole-filled layer while maintaining normal exciton states in the charge-neutral layer. The distinct resonant conditions of these layers effectively break interlayer inversion symmetry, thereby promoting resonant SHG. This method achieves a remarkable 40-fold enhancement of SHG at minimal electric field, equivalent to conditions near the dielectric-breakdown threshold but using only ∼3% of the critical breakdown field. Our results highlight SHG sensitivity to carrier density and type, offering a new tool for manipulating SHG and probing quantum states in 2D excitonic systems.

Self-selection of flow instabilities by acoustic perturbations around rectangular cylinder in cross-flow

This work experimentally investigates the flow structure around a rectangular cylinder with an aspect ratio of 2 under varying incidence angles to examine how acoustic perturbations modify and modulate the unsteady flow instabilities. In the absence of acoustic excitation, the angle of incidence is found to markedly influence the flow topology and the natural shedding pattern altering the vortex formation length and wake dynamics. With acoustic perturbations, it is observed that for incidence angles $\alpha = 0^{\circ }$ and $\alpha = 5^{\circ }$ , the masked impinging leading-edge vortex (ILEV)/trailing-edge vortex shedding (TEVS) instability modes of $n= 1$ and $n= 2$ become evident when their frequencies coincide with the frequencies of the acoustic perturbations (i.e. resonant condition). Both trailing-edge and leading-edge vortices were found to be modulated by the acoustic pressure cycle. These instability modes, which are naturally present under non-resonant conditions for significantly higher aspect ratios, highlight the role of incidence angle and self-excited acoustic resonance in virtually augmenting the streamwise length of the cylinder, thereby facilitating the emergence and sustenance of the ILEV/TEVS shedding pattern.

Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces

In this paper, we focus on the existence of response solutions, i.e. periodic solutions with the same frequencies as the external forces, for elliptic-hyperbolic partial differential equations with nonlinearities and periodic forces. The main tools are Lyapunov–Schmidt reduction and Nash–Moser iteration scheme, both of which have demonstrated success in hyperbolic scenarios. At each step of the iteration, the Galerkin approximation of the equation is solved. The new issue is that the spectral theory of the generalized Sturm–Liouville problem is employed, which also introduces new difficulties for estimations at each step. Under appropriate non-resonance conditions on the frequency, the existence of response solutions for the model will be established.

Non-Resonant Conditions for the Klein – Gordon Equation on the Circle

Non-Resonant Conditions for the Klein – Gordon Equation on the Circle

Effect of Reynolds Number and Aeroelastic Scaling Upon Launch-Vehicle Ground-Wind Loads

NASA conducted a launch vehicle ground-wind-loads investigation at the NASA Langley Transonic Dynamics Tunnel to investigate wind-induced oscillations (WIOs) of a launch vehicle when exposed to ground winds before launch. Previous publications from this effort have documented the effects of an atmospheric-boundary-layer profile on WIO response and the correlation process between model-scale and full-scale wind characteristics and resulting structural loads. This paper will focus on the importance of aeroelastic scaling and the impact of Reynolds number on WIO response. As described in the literature and confirmed in the present investigation, aeroelastic effects can significantly increase the magnitude of measured loads. Additionally, vortex shedding is sensitive to nuances of the flow in the shear layer, which is governed by Reynolds number. Many wind-tunnel facilities are not capable of producing flight Reynolds numbers for the ground-wind-loads problem. At very low Reynolds numbers, laminar shear layers exhibit different behavior, resulting in different vortex frequencies, oscillating lift magnitudes, and motion sensitivities. This investigation demonstrated that low Reynolds number testing can yield substantially lower dynamic loads with less aeroelastic coupling than those acquired at flight-representative Reynolds numbers for a resonant WIO event. Additionally, a resonant response phenomenon present at flight Reynolds number was absent at low Reynolds number. Conversely, for nonresonant WIO response conditions, similar dynamic load coefficients were obtained for similar test velocities at either Reynolds number condition. These findings impact many large launch vehicles, including the NASA Space Launch System series of vehicles.

Exponential convergence of the weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

Rotating solutions to the incompressible Euler–Poisson equation with external particle

We consider a two-dimensional, incompressible fluid body, together with self-induced interactions. The body is perturbed by an external particle with small mass. The whole configuration rotates uniformly around the common center of mass. We construct solutions, which are stationary in a rotating coordinate system, using perturbative methods. In addition, we consider a large class of internal motions of the fluid. The angular velocity is related to the position of the external particle and is chosen to satisfy a non-resonance condition.

Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems

Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic Hamiltonian system with zero equilibrium points for most small enough parameters. As an example, we discuss the reducibility and stability of an almost periodic Hill’s equation.

Fatigue Life Prediction and Verification of Railway Fastener Clip Based on Critical Plane Method

Accurately predicting the fatigue life of fastener clips is crucial for improving material processing technology, enhancing the anti-fatigue design level, and guiding the maintenance and repair of track structures. Conventional methods that solely evaluate the fatigue life of fastener clips based on the uniaxial stress index under displacement loads may lead to significant errors. In this paper, the stress field of a type-III clip under displacement loading conditions was numerically simulated based on fatigue test standards, and the fatigue life of the clip was analyzed using the Fatemi–Socie (FS) multiaxial fatigue criterion based on the critical plane method. A comparison with standard fatigue test results revealed that, under non-resonance conditions, the predicted position of the fatigue critical plane of the fastener clip coincided with the fracture surface observed at the middle of the measured small arc, with a life prediction error of 7.3%. To further investigate the predictive capability of the FS multiaxial fatigue criterion for the fatigue life of the fastener clip under resonance conditions, the stress levels of the clip under non-resonance and resonance conditions were compared through on-site testing, and the effects of inertial loads caused by vertical and lateral vibration acceleration on the stress field of the clip were analyzed in numerical simulation according to the results of clip acceleration tests; the fastener clip’s resonance fracture position and fatigue life were also predicted based on the aforementioned multiaxial fatigue criterion. To verify the accuracy of the predicted results, a testing method was proposed that equates the high-frequency resonant inertial load of the fastener clip to a low-frequency additional preload under the premise of consistent stress fields. A comparison with the numerical simulation results shows that considering only vertical inertial loads would result in a discrepancy between the measured fracture location and the actual one. Considering both vertical and lateral inertial loads, the fracture location (at the heel of the small arc) under resonance conditions could be accurately determined, with a life prediction error of 13.8%. Compared to the non-resonant displacement loading condition, the inertial loads caused by acceleration under resonance conditions led to a reduction in fatigue life of approximately 77.8%.

Energy Resolved Mass Spectrometry for Interoperable Non-resonant Collisional Spectra in Metabolomics.

In untargeted metabolomics, the unambiguous identification of metabolites remains a major challenge. This requires high-quality spectral libraries for reliable metabolite identification, which is essential for translating metabolomics data into meaningful biological information. Several attempts have been made to generate reproducible product ion spectra (PIS) under a low collision energy (ELab) regime and nonresonant collisional conditions but have not fully succeeded. We examined the ERMS (energy-resolved mass spectrometry) breakdown curves of two lipo-amino acids and showed the possibility to highlight "singular points", called descriptors hereafter (linked to respective ELab depending on the instrument), for each of the monomodal product ion profiles. Using several instruments based on different technologies, the PIS recorded at these specific ELab sites shows remarkable similarities. The descriptors appeared as being independent of the fragmentation mechanisms and can be used to overcome the main instrumental effects that limit the interoperability of spectral libraries. This proof-of-concept study, performed on two particular lipo-amino acids, demonstrates the high potential of ERMS-derived information to determine the instrument-specific ELab at which PIS recorded in nonresonant conditions become highly similar and instrument-independent, thus comparable across platforms. This innovative but straightforward approach could help remove some of the obstacles to metabolite identification in nontargeted metabolomics, putting an end to a challenging chimera.

Nonlinear model reduction to temporally aperiodic spectral submanifolds.

We extend the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems that are either weakly forced or slowly varying. Examples of such systems arise in structural dynamics, fluid-structure interactions, and control problems. The time-dependent SSMs we construct under these assumptions are normally hyperbolic and hence will persist for larger forcing and faster time dependence that are beyond the reach of our precise existence theory. For this reason, we also derive formal asymptotic expansions that, under explicitly verifiable nonresonance conditions, approximate SSMs and their aperiodic anchor trajectories accurately for stronger, faster, or even temporally discontinuous forcing. Reducing the dynamical system to these persisting SSMs provides a mathematically justified model- reduction technique for non-autonomous physical systems whose time dependence is moderate either in magnitude or speed. We illustrate the existence, persistence, and computation of temporally aperiodic SSMs in mechanical examples under chaotic forcing.

Quadratic Growth of Out-of-Time-Ordered Correlators in Quantum Kicked Rotor Model.

We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.

Coherent superposition of states in degenerate systems using zero-area pulses

The coherent superposition of states in degenerate quantum systems is investigated using detuned laser pulses in which the Rabi frequencies are time-dependent, and the pulse area is zero. In this study, a quantum system with an arbitrary number of degenerate states in the ground set as well as an arbitrary number of degenerate states in the exciting set is considered. We assume that all states in the ground set are coupled to the excited states using laser pulses such that the pulse area of Rabi frequency is zero, and all of them have the same time dependence. It is also assumed that all laser pulses are in a non-resonant condition with Bohr transitions, and all detunings are the same. We will show that by applying the appropriate temporal dependence for the pulses and the appropriate rate for the detunings, the population can be transferred from an arbitrary superposition from the ground states to a desired superposition from the excited states.

Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain’s Lemma which provides a partition of the “resonant sites” of the Laplace operator on irrational tori.

Existence of Solutions for an Impulsive p-laplacian Equation with Nonresonance Conditions

Existence of Solutions for an Impulsive p-laplacian Equation with Nonresonance Conditions

Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

On the local Gevrey integrability

This paper uses the Gevrey's smooth normalization theory to investigate the local integrability of vector fields, which have linear parts with one zero eigenvalue and the others non-resonant. First, we explore the general properties of C∞ local integrability. Then we show the same regularity of the Gevrey's smooth local integrability as that of the vector fields under Poincaré's non-resonant condition for the case that the real parts of the eigenvalues are all positive or negative. Lastly, a sharper expression of the loss of the regularity is provided by the lowest order of the resonant terms together with the indices of Gevrey's smoothness and the diophantine condition for the case that the matrix is in the diagonal form. One of the main tools utilized here is the KAM method in the Banach space fixed with a weighted norm.