Low-order Resonances Research Articles

Analytical study of influence of boundary conditions on acoustic power transfer through an elastic barrier

In this paper, a fully coupled electromechanical model of wireless power transfer through an elastic barrier using piezoelectric transducers is established based on the theory of piezoelectricity. In this model, an elastic layer representing the elastic barrier is sandwiched by two piezoelectric layers attached by boundary mass and boundary constraint. The two piezoelectric layers act as input energy converter and output energy receiver, respectively. Using the electrical and mechanical boundary and continuous conditions, all the electrical and mechanical components of the system are derived. A simplification is performed and compared with a previous study and a preliminary experiment to valid the present study. Numerical studies are conducted to reveal the effects of circuit load resistance, barrier thickness, boundary mass and boundary constraint on the resonance frequency, output voltage, output power and transmission efficiency of the present model. With different circuit load resistance, the system resonance frequency is found between the short-circuit resonance frequency and open-circuit resonance frequency. With boundary mass applied to the piezoelectric transducers, the maximum resonance frequency response of both the output power and the transmission efficiency move toward the low-order resonance frequency. With boundary constraint applied to the boundary mass, a newly generated resonance frequency with high transmission efficiency at the low-frequency range is observed. A method to recognize the necessary input frequency range as well as the necessary circuit load resistance range respectively for large output power and large transmission efficiency is proposed.

Realizing high transmission intensity in photonic crystal nanobeams using a side-coupling waveguide.

Side-coupled photonic crystal (PhC) nanobeam cavities were investigated to overcome challenges in measuring low-order resonances in traditional in-line PhC nanobeams that arise due to the trade-off between achieving high quality (Q)-factor and high transmission intensity resonances. On the same PhC nanobeam, we demonstrate that the side-coupling approach leads to measurable resonances even in cases in which high mirror strength unit cells severely limit the intensity of transmitted light through the in-line configuration. In addition, by coupling light directly into the cavity center, the design of side-coupled PhC nanobeams can be simplified such that high Q-factor PhC nanobeams can be achieved using only two different hole radii and uniform hole spacing.

Capture into first-order resonances and long-term stability of pairs of equal-mass planets

Massive planets form within the lifetime of protoplanetary disks and undergo orbital migration due to planet-disk interactions. When the first planet reaches the inner edge of the disk its migration stops and the second planet is locked in resonance. We detail the resonant trapping comparing semi-analytical formulae and numerical simulations in the case of two equal-mass coplanar planets trapped in first order resonances. We describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using expanded and non-expanded Hamiltonians. We show that during convergent migration the planets evolve along these families of equilibria. Eccentricity damping from the disk leads to a final equilibrium configuration that we predict analytically. The fact that observed multi-exoplanetary systems are rarely seen in resonances suggests that the resonant configurations achieved by migration can become unstable after the removal of the disk. We probe the stability of the resonances as a function of planetary mass. To do this, we fictitiously increase the masses of the planets, adiabatically maintaining the low-amplitude libration regime until instability. We discuss two hypotheses for instability: low-order secondary resonance of the libration frequency with a fast synodic frequency, and minimal approach distance between planets. We show that secondary resonances do not seem to impact resonant systems at low-amplitude of libration. Resonant systems are more stable than non-resonant ones for a given minimal distance at close encounters, but we show that the latter still play the decisive role in the destabilisation of resonant pairs. We show that, as the planetary mass increases and the minimal distance between planets gets smaller in terms of mutual Hill radius, the region of stability around the resonance center shrinks, until the equilibrium point itself becomes unstable.

Special Features of the Joint Influence of Low-Order Secular Resonances and Light Pressure on the Motion of Near-Earth Space Objects

Joint influence of secular resonances and light pressure on the long-term orbital evolution of objects in the near-earth space bounded by the semi-major axes of orbits from 15000 to 45000 km is considered. The orbital evolution of each object is considered for three area-to-mass ratios: 0.001, 1, and 10 m2/kg. Special attention is given to a search for resonances whose influence increases the eccentricities of near-earth space object orbits and to their joint analysis with the influence of the light pressure force on the object dynamics.

Interaction of Lower and Higher Order Hamiltonian Resonances

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. Applications are given to the three dof [Formula: see text] resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.

Mathematical Simulation of Perturbations of Attack Angle of Asymmetric Nanosatellite Passing through Resonance

The authors consider the problem of a dynamic system passing through a low-order resonance, describing an uncontrolled atmospheric descent of an asymmetric nanosatellite in the Earth's atmosphere. The authors perform mathematical and numerical modeling of the motion of the nanosatellite with a small mass-aerodynamic asymmetry relative to the center of mass. The aim of the study is to obtain new reliable approximate analytical estimates of perturbations of the angle of attack of a nanosatellite passing through resonance at angles of attack of not more than 0.5π. By using the stationary phase method, the authors were able to investigate a discontinuous perturbation in the angle of attack of a nanosatellite passing through a resonance with two different nanosatellite designs. Comparison of the results of the numerical modeling and new approximate analytical estimates of the perturbation of the angle of attack confirms the reliability of the said estimates.

Vibration due to non-circularity of a rotating ring having discrete radial supports - With application to thin-walled rotor/magnetic bearing systems

This paper investigates the vibration arising in a thin-walled cylindrical rotor subject to small non-circularity and coupled to discrete space-fixed radial bearing supports. A Fourier series description of rotor non-circularity is incorporated within a mathematical model for vibration of a rotating annulus. This model predicts the multi-harmonic excitation of the rotor wall due to bearing interactions. For each non-circularity harmonic there is a set of distinct critical speeds at which resonance can potentially arise due to flexural mode excitation within the rotor wall. It is shown that whether each potential resonance occurs depends on the multiplicity and symmetry of the bearing supports. Also, a sufficient number of evenly spaced identical supports will eliminate low order resonances. The considered problem is pertinent to the design and operation of thin-walled rotors with active magnetic bearing (AMB) supports, for which small clearances exist between the rotor and bearing and so vibration excitation must be limited to avoid contacts. With this motivation, the mathematical model is further developed for the case of a distributed array of electromagnetic actuators controlled by feedback of measured rotor wall displacements. A case study involving an experimental system with short cylindrical rotor and a single radial AMB support is presented. The results show that flexural mode resonance is largely avoided for the considered design topology. Moreover, numerical predictions based on measured non-circularity show good agreement with measurements of rotor wall vibration, thereby confirming the validity and utility of the theoretical model.

Hierarchical collapse of regular islands via dissipation

In this work we investigate how regular islands localized in a mixed phase-space of generic area-preserving Hamiltonian systems are affected by a small amount of dissipation. Mainly we search for a universality (hierarchy) in the convergence of higher-order resonances and their periods when dissipation increases. One very simple scenario is already known: when subjected to small dissipation, stable periodic points become sinks attracting almost all the surrounding orbits, destroying all invariant curves which divide the phase-space in chaotic and regular domains. However, performing numerical experiments with the paradigmatic Chirikov–Taylor standard mapping we show that this presumably simple scenario can be rather complicated. The first, not trivial, scenario is what happens to chaotic trajectories, since they can be attracted by the sinks or by chaotic attractors, in cases when they exist. We show that this depends very much on how basins of attraction are formed as dissipation increases. In addition, we demonstrate that higher-order resonances are usually first affected by small dissipation when compared to lower-order resonances from the conservative case. Nevertheless, this is not a generic behaviour. We show that a local hierarchical collapse of resonances, as dissipation increases, is related to the area of the islands from the conservative case surrounding the periodic orbits. All observed resonance destructions occur via the bifurcation phenomena and are quantified here by determining the largest finite-time Lyapunov exponent.

A compact metamaterial loaded monopole antenna with offset-fed microstrip line for wireless applications

A compact metamaterial loaded monopole antenna with offset-fed microstrip line is proposed for Universal Mobile Telecommunication System (UMTS), Worldwide interoperability for Microwave Access (WiMAX), and Wireless Local Area Network (WLAN) wireless applications. The proposed antenna is printed on a 19.18×22.64×1.6mm3 FR-4 substrate having a dielectric constant (εr) of 4.4. The antenna radiating element consists of split ring structure and CSRR for generating multiband characteristics, which is fed by offset microstrip line. A split in the outer vertical arm creates a lower order resonance at 2.1GHz and the Complementary Split Ring Resonator (CSRR) in the monopole antenna is used to generate a new resonance frequency of 3.45GHz. The precise equivalent circuit design equations are used to analyze the CSRR resonance frequency. Also, the band characteristics of a split ring structure and CSRR are enlightened in detail to verify the metamaterial property. Simulated results are verified with measured results. The measured azimuthal plane (H-Plane) exhibits omnidirectional radiation pattern and elevation plane (E-plane) represents a bidirectional radiation pattern.

Numerical Simulation and Experimental Validation of Failure Caused by Vibration of a Fan

This paper presents the root cause analysis of an unexpected fracture occurred on the blades of a motor fan used in a natural gas reciprocating compressor unit. A finite element model was established to investigate the natural frequencies and modal shapes of the fan, and a modal test was performed to verify the numerical results. It was indicated that the numerical results agreed well with experimental data. The third order natural frequency was close to the six times excitation frequency, and the corresponding modal shape was the combination of bending and torsional vibration, which consequently contributed to low-order resonance and fracture failure of the fan. The torsional moment obtained by a torsional vibration analysis of the compressor shaft system was exerted on the numerical model of the fan to evaluate the dynamic stress response of the fan. The results showed that the stress concentration regions on the numerical model were consistent with the location of fractures on the fan. Based on the numerical simulation and experimental validation, some recommendations were given to improve the reliability of the motor fan.

Coherent resonance stop bands in alternating gradient beam transport

An extensive experimental study is performed to confirm fundamental resonance bands of an intense hadron beam propagating through an alternating gradient linear transport channel. The present work focuses on the most common lattice geometry called ``FODO'' or ``doublet'' that consists of two quadrupoles of opposite polarities. The tabletop ion-trap system ``S-POD'' (Simulator of Particle Orbit Dynamics) developed at Hiroshima University is employed to clarify the parameter-dependence of coherent beam instability. S-POD can provide a non-neutral plasma physically equivalent to a charged-particle beam in a periodic focusing potential. In contrast with conventional experimental approaches relying on large-scale machines, it is straightforward in S-POD to control the doublet geometry characterized by the quadrupole filling factor and drift-space ratio. We verify that the resonance feature does not essentially change depending on these geometric factors. A few clear stop bands of low-order resonances always appear in the same pattern as previously found with the sinusoidal focusing model. All stop bands become widened and shift to the higher-tune side as the beam density is increased. In the space-charge-dominated regime, the most dangerous stop band is located at the bare betatron phase advance slightly above 90 degrees. Experimental data from S-POD suggest that this severe resonance is driven mainly by the linear self-field potential rather than by nonlinear external imperfections and, therefore, unavoidable at high beam density. The instability of the third-order coherent mode generates relatively weak but noticeable stop bands near the phase advances of 60 and 120 degrees. The latter sextupole stop band is considerably enhanced by lattice imperfections. In a strongly asymmetric focusing channel, extra attention may have to be paid to some coupling resonance lines induced by the Coulomb potential. Our interpretations of experimental data are supported by theoretical predictions and systematic multiparticle simulations.

Escape of asteroids from the main belt

Aims. We locate escape routes from the main asteroid belt, particularly into the near-Earth-object (NEO) region, and estimate the relative fluxes for different escape routes as a function of object size under the influence of the Yarkovsky semimajor-axis drift.Methods. We integrated the orbits of 78 355 known and 14 094 cloned main-belt objects and Cybele and Hilda asteroids (hereafter collectively called MBOs) for 100 Myr and recorded the characteristics of the escaping objects. The selected sample of MBOs with perihelion distance q > 1.3 au and semimajor axis a V V We find more than ten obvious escape routes from the asteroid belt to the NEO region, and they typically coincide with low-order mean-motion resonances with Jupiter and secular resonances. The locations of the escape routes are independent of the semimajor-axis drift rate and thus are also independent of the asteroid diameter. The locations of the escape routes are likewise unaffected when we added a model for Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) cycles coupled with secular evolution of the rotation pole as a result of the solar gravitational torque. A Yarkovsky-only model predicts a flux of asteroids entering the NEO region that is too high compared to the observationally constrained flux, and the discrepancy grows larger for smaller asteroids. A combined Yarkovsky and YORP model predicts a flux of small NEOs that is approximately a factor of 5 too low compared to an observationally constrained estimate. This suggests that the characteristic timescale of the YORP cycle is longer than our canonical YORP model predicts.

Importance of transient resonances in extreme-mass-ratio inspirals

The inspiral of stellar-mass compact objects, like neutron stars or stellar-mass black holes, into supermassive black holes provides a wealth of information about the strong gravitational-field regime via the emission of gravitational waves. In order to detect and analyse these signals, accurate waveform templates which include the effects of the compact object's gravitational self-force are required. For computational efficiency, adiabatic templates are often used. These accurately reproduce orbit-averaged trajectories arising from the first-order self-force, but neglect other effects, such as transient resonances, where the radial and poloidal fundamental frequencies become commensurate. During such resonances the flux of gravitational waves can be diminished or enhanced, leading to a shift in the compact object's trajectory and the phase of the waveform. We present an evolution scheme for studying the effects of transient resonances and apply this to an astrophysically motivated population. We find that a large proportion of systems encounter a low-order resonance in the later stages of inspiral; however, the resulting effect on signal-to-noise recovery is small as a consequence of the low eccentricity of the inspirals. Neglecting the effects of transient resonances leads to a loss of 4% of detectable signals.

A simple model for the location of Saturn’s F ring

In this paper, we introduce a simplified model to understand the location of Saturn’s F ring. The model is a planar restricted five-body problem defined by the gravitational field of Saturn, including its second zonal harmonic J2, the shepherd moons Prometheus and Pandora, and Titan. We compute accurate long-time numerical integrations of (about 2.5 million) non-interacting test-particles initially located in the region between the orbits of Prometheus and Pandora, and address whether they escape or remain trapped in this region. We obtain a wide region of initial conditions of the test particles that remain confined. We consider a dynamical stability indicator for the test particles’ motion defined by computing the ratio of the standard deviation over the average value of relevant dynamical quantities, in particular, for the mean-motion and the semi-major axis. This indicator separates clearly a subset of trapped initial conditions that appear as very localized stripes in the initial semi-major axis and eccentricity space for the most stable orbits. Retaining only these test particles, we obtain a narrow eccentric ring which displays sharp edges and collective alignment. The semi-major axis of the accumulation stripes of the stable ring-particles can be associated with resonances, mostly involving Prometheus’ outer Lindblad and co-rotation resonances, but not exclusively. Pandora’s inner Lindblad and co-rotation resonances as well as low-order three-body resonances typically coincide with gaps, i.e., regions of instabilities. Comparison of our results with the nominal data for the F ring shows some correspondence.

Fabrication and optical measurement of double-overlapped annular apertures

We demonstrate double-overlapped annular aperture (DOAA) arrays fabricated in a gold film via focused ion beam milling. The high order resonance modes of DOAA are investigated both theoretically and experimentally. Polarization dependency is observed for DOAA arrays and lower order resonance modes in the mid-infrared range exhibit extraordinary optical transmission and dependency on geometric parameters.

The effects of bar–spiral coupling on stellar kinematics in the Galaxy

We investigate models of the Milky Way disc taking into account simultaneously the bar and a two-armed quasi-static spiral pattern. Away from major resonance overlaps, the mean stellar radial motions in the plane are essentially a linear superposition of the isolated effects of the bar and spirals. Thus, provided the bar is strong enough, even in the presence of spiral arms, these mean radial motions are predominantly affected by the Galactic bar for large scale velocity fluctuations. This is evident when comparing the peculiar line-of-sight velocity power spectrum of our coupled models with bar-only models. However, we show how forthcoming spectroscopic surveys could disentangle bar-only non-axisymmetric models of the Galaxy from models in which spiral arms have a significant amplitude. We also point out that overlaps of low-order resonances are sufficient to enhance stellar churning within the disc, even when the spirals amplitude is kept constant. Nevertheless, for churning to be truly non-local, stronger or (more likely) transient amplitudes would be needed: otherwise the disc is actually mostly unaffected by churning in the present models. Finally, regarding vertical breathing modes, the combined effect of the bar and spirals on vertical motions is a clear non-linear superposition of the isolated effects of both components, significantly superseding the linear superposition of modes produced by each perturber separately, thereby providing an additional effect to consider when analysing the observed breathing mode of the Galactic disc in the extended Solar neighbourhood.

Numerical and experimental investigation of the torsional stiffness of flexible disc couplings

Disc couplings are widely used in compressors, gas turbines, and aerospace applications because of their flexibility and their consequent ability to compensate in almost all directions. The torsional stiffness of disc coupling has a great influence on shaft torsional vibration, and in many applications low-order torsional resonance may occur due to insufficient stiffness in the disc coupling. Investigation of the factors influencing the torsional stiffness of disc coupling will help to improve the design and control of shaft torsional vibration. In this paper, a 3D finite element (FE) model was built to estimate the stiffness of a disc coupling, taking the behavior of friction and contact into consideration. Based on this model, the curve of torque vs. angular displacement was obtained by applying a series of torsional load steps to the coupling. The effects of diaphragm shape, bolt preload, and fluctuation of torque were evaluated. A test rig was established to validate the simulated results. Both the simulated and experimental results demonstrated a strong nonlinear relationship between the torque and angular displacement during both the loading and unloading processes. The results also showed that the bending and inclining of flanges and diaphragms were mainly responsible for the varied stiffness of the disc coupling. In addition, load fluctuation, bolt preload, and shape of diaphragms could lead to variation in the stiffness of disc couplings.

Spin-up of massive classical bulges during secular evolution

Classical bulges in spiral galaxies are known to rotate but the origin of this observed rotational motion is not well understood. It has been shown recently that a low-mass classical bulge (ClB) in a barred galaxy can acquire rotation from absorbing a significant fraction of the angular momentum emitted by the bar. Our aim here is to investigate whether bars can spin up also more massive ClBs during the secular evolution of the bar, and to study the kinematics and dynamics of these ClBs. We use a set of self-consistent N-body simulations to study the interaction of ClBs with a bar that forms self-consistently in the disk. We use orbital spectral analysis to investigate the angular momentum gain by the classical bulge stars. We show that the ClBs gain, on average, about 2 - 6% of the disk's initial angular momentum within the bar region. Most of this angular momentum gain occurs via low-order resonances, particularly 5:2 resonant orbits. A density wake forms in the ClB which corotates and aligns with the bar at the end of the evolution. The spin-up process creates a characteristic linear rotation profile and mild tangential anisotropy in the ClB. The induced rotation is small in the centre but significant beyond $\sim2$ bulge half mass radii, where it leads to mass-weighted $V/\sigma \sim 0.2$, and reaches a local $V_{max}/\sigma_{in} \sim 0.5$ at around the scale of the bar. The resulting $V/\sigma$ is tightly correlated with the ratio of the bulge size to the bar size. In all models, a box/peanut bulge forms suggesting that composite bulges may be common. Bar-bulge resonant interaction in barred galaxies can provide some spin up of massive ClBs, but the process appears to be less efficient than for low-mass ClBs. Further angular momentum transfer due to nuclear bars or gas inflow would be required to explain the observed rotation if it is not primordial.

The resonance motions of a statically stable Lagrange top at small nutation angles

The resonance motion modes of a Lagrange top with small mass asymmetry upon passage through low-order resonances are analysed. The motion of a rigid body with a prolate ellipsoid of inertia is considered in the vicinity of a lower, statically stable equilibrium position at small nutation angles. The original system of equations is reduced to the equation of motion of a pendulum with small perturbations using an averaging method. The conditions for plunging the phase trajectory of the system into oscillation regions of the pendulum, i.e., the possibility of capture into resonance under the action of the perturbations considered, are analysed. Cases in which captures into resonance are possible and in which they are not are established. Numerical calculations that illustrate the analytical estimates obtained are presented.

Quantifying the Sensitivity of Multipolar (Dipolar, Quadrupolar, and Octapolar) Surface Plasmon Resonances in Silver Nanoparticles: The Effect of Size, Composition, and Surface Coating.

The effect of composition, size, and surface coating on the sensitivity of localized multipolar surface plasmon resonances has been spectroscopically investigated in high-quality silver colloidal solutions with precisely controlled sizes from 10 to 220 nm and well-defined surface chemistry. Surface plasmon resonance modes have been intensively characterized, identifying the size-dependence of dipolar, quadrupolar, and octapolar modes. Modifications of the NP's surface chemistry revealed the higher sensitivity of large sizes, long molecules, thiol groups, and low-order resonance modes. We also extend this study to gold nanoparticles, aiming to compare the sensitivity of both materials, quantifying the higher sensitivity of silver.