Order Resonance Research Articles

Vibroacoustic Transfer Characteristics of Underwater Cylindrical Shells Containing Complex Internal Elastic Coupled Systems

Cylindrical shells containing complex elastic coupling systems are the main structural form of underwater vehicles. Therefore, in this paper, the vibroacoustic radiation problem of underwater cylindrical shells containing complex internal elastic coupling systems is studied. Firstly, the dynamics model of the complex elastic coupled system is established through the method of integrated conductivity. The sound pressure distribution law and the general magnitude relationship between the performance index of hydroacoustic radiation and vibration isolation are investigated through numerical simulation. A strategy of global sensitivity analysis and related parameter optimization is carried out, by applying the Sobol’ method to the dynamics model. It could be concluded that the main flap of sound pressure at low and medium frequencies appears in the direction of the excitation force or the perpendicular to the excitation force, the magnitudes correspondence between the vibration level drop—power flow—hydroacoustic radiation at low frequencies can be expressed as a relatively simple function, and the vibroacoustic transmission of the system at lower order resonance frequencies is dominated by the parameter configuration of the vibration isolation device, while at higher frequencies is more influenced by the modalities of the base structure. The transfer power flow and the level drop are used as objective functions to optimise the acoustic radiation index of the coupled system, with the best results obtained when the transfer power flow and the level drop are used together as objective functions.

Responses of a Strongly Forced Mathieu Equation—Part 1: Cyclic Loading

AbstractThis work concerns the response of a damped Mathieu equation with hard cyclic excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales unfolds resonances and stability. Superharmonic and subharmonic resonances are analyzed and the effect of different parameters on the responses are examined. While superharmonic resonances of order two have been captured by a first-order analysis, the second-order analysis improves the prediction of the peak frequency. Superharmonic resonances of order three are captured only by the second-order analysis. The order-two superharmonic resonance amplitude is of order ε0, and the order-three superharmonic amplitude is of order ε. As the parametric excitation level increases, the superharmonic resonance amplitudes increase. An nth-order multiple-scales analysis will indicate conditions of superharmonic resonances of order n + 1. At the subharmonic of order one-half, there is no steady-state resonance, but known subharmonic instability is unfolded consistently. Analytical expressions for resonant responses are presented and compared with numerical results for specific system parameters. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric resonators.

Responses of a Strongly Forced Mathieu Equation—Part 2: Constant Loading

AbstractThe present study deals with the response of a damped Mathieu equation with hard constant external loading. A second-order perturbation analysis using the method of multiple scales (MMS) unfolds resonances and stability. Non-resonant and low-frequency quasi-static responses are examined. Under constant loading, primary resonances are captured with a first-order analysis, but are accurately described with the second-order analysis. The response magnitude is of order ϵ0, where ϵ is the small bookkeeping parameter, but can become arbitrarily large due to a small denominator as the Mathieu system approaches the primary instability wedge. A superharmonic resonance of order two is unfolded with the second-order MMS. The magnitude of this response is of order ϵ and grows with the strength of parametric excitation squared. An nth-order multiple scales analysis under hard constant loading will indicate conditions of superharmonic resonances of order n. Subharmonic resonances do not produce a non-zero steady-state harmonic, but have the instability property known to the regular Mathieu equation. Analytical expressions for predicting the magnitude of responses are presented and compared with numerical results for a specific set of system parameters. In all cases, the second-order analysis accommodates slow time-scale effects, which enable responses of order ϵ or ϵ0. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric amplifiers.

Non-perturbative investigation of low-eccentricity exterior mean motion resonances

ABSTRACT Mean motion resonances are important in the analysis and understanding of the dynamics of planetary systems. While perturbative approaches have been dominant in many previous studies, recent non-perturbative approaches have revealed novel properties in the low-eccentricity regime for interior mean motion resonances of Jupiter in the fundamental model of the circular planar restricted three-body model. Here, we extend the non-perturbative investigation to exterior mean motion resonances in the low-eccentricity regime (up to about 0.1) and for perturber mass in the range of ∼5 × 10−5 to 1 × 10−3 (in units of the central mass). Our results demonstrate that first-order exterior resonances have two branches at low eccentricity as well as low-eccentricity bridges connecting neighbouring first-order resonances. With increasing perturber mass, higher order resonances dissolve into chaos, whereas low-order resonances persist with larger widths in their radial extent but smaller azimuthal widths. For low-order resonances, we also detect secondary resonances arising from small-integer commensurabilities between resonant librations and the synodic frequency. These secondary resonances contribute significantly to generating the chaotic sea that typically occurs near mean motion resonances of higher mass perturbers.

A Novel Compact Filtering Antenna for 5.0-GHz WLAN Communication System

The proposed antenna presents a microstrip filter antenna with both filter out-of-band rejection characteristics and antenna radiation characteristics for WLAN communication system. The antenna consists of four U-shaped microstrip resonators, one Γ-shaped antenna, and one parallel coupling line. The Γ-shaped antenna has the filtering function as the last order resonator, and the parallel coupling line integrates the filter and antenna as the conductance converter. After simulation and test, the results show that the center frequency of the filter antenna is 5.25 GHz, the fractional bandwidthis 8.0% (5.0 − 5.42 GHz), the in-band gain changes smoothly, the maximum in-band gain is 3.059 dBi, the gain at 150 MHz outside the passband rapidly decreases to 10.0 dBi, and the overall out-of-band gain drops to below −24 dBi. The filter antenna has a simple structure and thin substrate, which optimizes the overall size of the filter and antenna, reduces the transmission loss between them, and is applicable to the RF front-end of wireless communication systems.

Highly Selective Broadband mm-Wave Diplexer Design

This letter presents a novel broadband diplexer design that is capable of achieving highly stringent rejection characteristics through the use of singlets whose interconnecting irises are formulated as resonant slot irises. The combination of these two resonant-cavity types allows for a unique filtering solution with increased filter order, wide available bandwidth, low geometric complexity, and simple milling requirements, which can be suitably applied to millimeter-wave and submillimeter-wave applications. A prototype is fabricated for operation in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> -band (75–110 GHz) in order to cover a 10% fractional bandwidth in each passband. Measurement of the prototype denotes highly accurate results and exemplifies the use of all resonator and coupling elements in order to support ten poles and four transmission zeros in an elegant diplexer solution.

Design Procedure for Reduced Filter Size in a Buck Converter Using a Fourth-Order Resonance Filter

This article presents a novel design procedure for fourth order and fourth-order resonance (4thRes) output filters, for given buck converter specifications, making components selection a straightforward process. An accurate filter analysis is provided to predict the filter component currents and voltages in both frequency and time domains. Application of the analysis in a design study of a 20 MHz, 5.4 W buck converter shows that the 4thRes filter has the potential to reduce the output passive components for a wide duty cycle range. As compared with a second-order filter at <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IN</sub> = 6.6 V to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">OUT</sub> = 1.8 V, total inductance, inductor energy, capacitance, and capacitor energy are 58%, 35%, 45%, and 31% lower, respectively. Air-core printed circuit board (PCB) integrated solenoid inductors are considered for implementation and testing within a prototype converter to show the impact of these filters on the converter performance. The 4thRes filter achieved 3.7% and 3.6% higher full-load efficiency than the second- and fourth-order filters, respectively, and a better load transient performance.

Символьное вычисление условия резонанса произвольного порядка в системе Гамильтона

The study of formal stability of equilibrium positions of a multiparametric Hamiltonian system in a generic case is traditionally carried out using its normal form under the condition of the absence of resonances of small orders. In this paper we propose a method of symbolic computation of the condition of existence of a resonance of arbitrary order for a system with three degrees of freedom. It is shown that this condition for each resonant vector can be represented as a rational algebraic curve. By methods of computer algebra the rational parametrization of this curve for the case of an arbitrary resonance is obtained. A model example of some two-parameter system of pendulum type is considered.

Arbitrary variable thickness annular piezoelectric ultrasonic transducer based on transfer matrix method

The variable thickness annular radial piezoelectric ultrasonic transducer can realize impedance transformation and energy concentration, has the advantages of large radiation area and full directivity, and is widely used in power ultrasound, underwater acoustic and other fields. Because solving complex variable thickness metal ring radial vibration wave equation is more difficult, in this paper, the radial vibration of metal rings with variable thickness is transformed into the superposition of the radial vibrations of N metal rings with equal thickness by using the transfer matrix method. The equivalent circuit diagram, the resonance frequency equation and the expression of the displacement amplification coefficient of the radial vibration of the metal thin ring with arbitrary thickness are obtained. The relationship between the displacement amplification coefficient and the geometric size of the cone, power function, exponential and catenary metal rings is analyzed. On this basis, the equivalent circuit and resonance frequency equation of radial vibration of piezoelectric ultrasonic transducer which is composed of a metal ring with variable thickness and a piezoelectric ring with equal thickness are derived. In order to verify the correctness of the theoretical results, the finite element software is used in simulation, and the numerical solutions of the first and second order resonance frequency and displacement amplification coefficients are in good agreement with the theoretical solutions. In this paper, the universal solution of radial vibration of metal ring with arbitrary variable thickness is given, which provides theoretical guidance for designing and optimizing the radial piezoelectric ultrasonic transducers.

Algorithmic Signal Accuracy Improvement of Solid-State Wave Gyroscopes

Possible ways to algorithmically improve the accuracy of the output signals of solid-state wave gyroscopes of reduced manufacturing quality are discussed - if their resonators have residual mechanical defects in the form of anisotropy of the elastic and damping properties. For this purpose, narrowband amplitude-phase models of oscillations of resonators at the second angular harmonic, recorded in resonant variables, were used. Next, the transition to arbitrary fixed axes, including the instrument axes of the measuring device, is made. A transition to the movable axes of the working and quadrature standing waves was also made to reveal parametric dependencies in the drift function of the output signal of the gyroscope. On this basis, seven main stages of sequential system algorithmic improvement of output signal accuracy are outlined. Among them, the article mainly considers options for combining calibration operations for systematic drift functions of the output signal with new opportunities to improve the accuracy of the signal calculation algorithm. The highlighted options are: the plotting of two-dimensional calibration functions; application of theoretical relations to reduce the complexity of calibration of the drift function; separation of drift functions into calibrated and measured parts. To calibrate the coefficients of the theoretical parts of the drift functions, different models of their identification in the mode of free oscillations of the gyroscope resonator are considered. At the same time, models of oscillations in the movable axes of standing waves and fixed instrument axes were used. Additionally, for the case of a single-electrode parametric excitation system, the possibilities of controlling anisotropy of the elastic and damping properties of gyroscope resonators in the operating mode of exploitation are briefly indicated. The task of identifying the parameters of reference models of oscillations of resonators in order to form alternative algorithms for improving the accuracy of the measurement signal of gyroscopes was also set. The development of the described areas of search for ways to algorithmically improve the accuracy of the signal of solid-state wave gyroscopes with significant residual processing errors of their resonators in practice is expected in the process of their direct development in specific operating conditions.

Ultrasonic interference and critical attenuation in metal-plastic bilayer laminates

The resonance of ultrasonic waves in a metal-plastic bilayer laminate placed in an infinite medium is theoretically investigated, and critical attenuation is shown to occur under specific conditions. The bilayer laminate is subjected to normal wave incidence from the side of the plastic layer. Based on a linear viscoelastic model, the amplitude of the reflection spectrum for the laminate is formulated to obtain the exact results for wave resonance and critical attenuation by numerical calculations. Furthermore, the approximate formulae for the conditions of the resonance and critical attenuation are derived explicitly by Taylor expansions with respect to the loss factor of the plastic. As a result, the exact solution shows that the resonance frequencies of the plastic layer slightly increase with increasing loss factor. Their variation with the loss factor agrees well with the second-order approximation result but is insignificant in the cases of common plastics with relatively low loss factors. The exact results also show that the amplitude at each order resonance depends on the loss factor and has a minimum at a critical loss factor. The obtained critical loss factor decreases with increasing resonance order and is well reproduced by the zeroth-order approximation. Experiments are performed on two types of metal-plastic bonded laminates, demonstrating that the measured reflection spectra are in good agreement with the theoretical predictions. In particular, the amplitude of the measured reflection spectrum decreases significantly at a resonance frequency, which is theoretically predicted to satisfy the critical attenuation condition.

Amplitude–frequency response of parametric resonance of electrostatically actuated MEMS clamped circular plate

This paper investigates the amplitude–frequency response of parametric resonance of a clamped elastic circular plate microelectromechanical system (MEMS) resonator above a parallel ground plate and under electrostatic actuation. Soft AC voltage with frequency near the first natural frequency of the plate is used. This results into parametric resonance. The system is assumed to be weakly nonlinear. Numerical and analytical methods are used to solve the reduced order models of the electrostatically actuated MEMS circular plate resonators in order to obtain the amplitude–frequency response of their parametric resonance. Seven Reduced Order Models (ROM) with one to seven modes of vibration (terms) have been developed and used in this investigation. ROM with one mode of vibration was solved using the Method of Multiple Scales (MMS) and predicted the existence of the resonance. ROMs with two to seven modes of vibration were solved using continuation and bifurcation software AUTO 07p to simulate and predict the amplitude–frequency response. These simulations showed that increasing the number of modes of vibration in the ROM produced better results. However, there is no significant difference between six and seven modes of vibration ROMs. Therefore, the ROM using seven modes of vibration was used in this research. This ROM was also numerically integrated to predict time responses of the MEMS plate. All methods showed an excellent agreement for amplitudes less than 0.5 of the gap. Only ROM using seven modes of vibrations had accurate predictions for all amplitudes, i.e. amplitudes between zero and the gap distance. The frequency response consists of two bifurcations, subcritical and supercritical. The effects of AC voltage and damping on the amplitude–frequency response are reported. Increasing the AC voltage results in shifting of the bifurcations to lower frequencies, subcritical significantly more than supercritical. Increasing the damping results in a narrower frequency range between the bifurcations.

High-resolution resonant portraits of a single-planet white dwarf system

ABSTRACT The dynamical excitation of asteroids due to mean motion resonant interactions with planets is enhanced when their parent star leaves the main sequence. However, numerical investigation of resonant outcomes within post-main-sequence simulations is computationally expensive, limiting the extent to which detailed resonant analyses have been performed. Here, we combine the use of a high-performance computer cluster and the general semi-analytical libration width formulation of Gallardo, Beaugé &amp; Giuppone in order to quantify resonant stability, strength, and variation instigated by stellar evolution for a single-planet system containing asteroids on both crossing and non-crossing orbits. We find that resonant instability can be accurately bound with only main-sequence values by computing a maximum libration width as a function of asteroid longitude of pericentre. We also quantify the relative efficiency of mean motion resonances of different orders to stabilize versus destabilize asteroid orbits during both the giant branch and white dwarf phases. The 4:1, 3:1, and 2:1 resonances represent efficient polluters of white dwarfs, and even when in the orbit-crossing regime, both the 4:3 and 3:2 resonances can retain small reservoirs of asteroids in stable orbits throughout giant branch and white dwarf evolution. This investigation represents a preliminary step in characterizing how simplified extrasolar Kirkwood gap structures evolve beyond the main sequence.

Nonlinear vibration of the axle box front cover of high-speed train and its effect on connecting bolts

In order to explore the causes for failure of connecting bolts between the axle box front cover and axle box body on high-speed train, field test was carried out firstly in this work. The results showed that the wheel had 20th-order polygonization and the front cover vibrated drastically with the dominant frequencies of 290 Hz and 1160 Hz at the speed of 300 km/h, which were 1/2 and 2 times the rotational frequency 580 Hz of 20th-order polygonal wheel, respectively. Secondly, bench test was carried out to reproduce the unique and drastic vibration phenomenon of the field test, it was proved that the wheel polygonization took responsibility for the drastic vibration of the front cover. OMA (Operational Modal Analysis) and theoretical analysis were made next to certify existence of nonlinear vibration (subharmonic resonance of order 1/2 and superharmonic resonance of order 2) of the front cover under 20th-order polygonization. Finally, the cause of bolt failure and dynamic characteristic of the front cover resulted from different wheel polygon order was studied, results showed that relative vibration existed between axle box body and front cover. The relative vibration enlarged when the front cover was in nonlinear vibration under 20th-order wheel polygonization, which definitely caused a larger dynamic transverse load for the bolts and accelerated its vibration fatigue.

Terahertz ultra-wideband antenna with tunable band-notch creation using resonant/non-resonant graphene loop

The state of graphene loop applied in the slotted antenna radiator is converted from non-resonant to resonant to achieve the tunable band-notch characteristics in an ultra-wideband (UWB) antenna. In the non-resonant state, the graphene loop with value of chemical potential as 0 eV acts as a parasitic element and does not contribute to radiation. Increasing the chemical potential of graphene starts converting the state of graphene loop from non-resonant to resonant. The graphene surface starts confining the localized surface plasmons with the increment in chemical potential and hence the multiple higher order resonances are formed. The higher order resonance with the destructive effect provide the band-notch characteristic in UWB response. The variation in chemical potential can tune the created notched band over frequency. Also, conversion of state of graphene loop from non-resonant to resonant deteriorates the far-field radiation parameters and radiation efficiency of antenna drastically. A trade-off can be set to find the required value of gain and radiation efficiency and frequency of the created band notch. Moreover, another technique of utilizing the double graphene loops with less surface area has also been studied, which allows one to attain the higher radiation efficiency of antenna along with the band-notch characteristics.

Excitation of higher order modes in terahertz antenna using graphene strip

A terahertz (THz) antenna is implemented with a technique of achieving the multiband response by exciting the higher order modes without changing the antenna structure. A patterned triangular shaped THz microstrip patch radiator is designed and numerically analyzed for providing the dual-band response with TM10 and TM20 modes. A graphene strip is applied in the antenna structure which can act as a controller or tuner responsible for the generation and mitigation of resonant modes. The insertion of graphene strip can generate the higher order resonances of TM11 and TM21 mode and antenna response can be attained with dual, triple or four operating passbands by varying the chemical potential of the applied graphene strip. The antenna operation is verified with the electrical equivalent circuit model. Moreover, conical or bi-conical radiation beam can be obtained from the antenna corresponding to the nature of the excited resonant mode.

Tuning the higher to lower order resonance frequency ratio and implementing the tunable THz MIMO/self-diplexing antenna

The frequency ratio of higher to lower order mode can be electronically tuned in a terahertz (THz) antenna with metallic radiator using a graphene loop. Antenna is designed with slotted metallic radiator to obtain the dual-band response with fundamental and second order transverse magnetic mode providing the directional and bi-directional radiation pattern in the lower and upper band, respectively. The insertion of graphene loop and varying its chemical potential (μc) provides two resonances until 0.4eV and four resonances for further greater values of μc. The desired impedance matching can be achieved at the frequencies of any two resonances at a time by selecting an appropriate value of μc. Antenna can operate with higher/lower order mode centred at frequency 3.77/3.02 THz and 4.39/2.86 THz for the values of μc as 0.9 and 0.3eV , respectively. The frequency ratio of higher to lower order mode can be tuned within the range of 1.22–1.59 with the variation in μc. Also, application of graphene loop confines the radiated power in a single direction at the frequency of higher order mode making the radiation pattern consistent. Antenna can be utilized in future THz wireless applications which require the utilization of adjacent channels with different frequencies in communication. Also, a two-port antenna is designed which can offer the tunable multi-input–multi-output (MIMO) and self-diplexing capability with pattern diversity. The MIMO parameters; envelope correlation coefficient (ECC) and diversity gain (DG) is evaluated and their values are found within acceptable limits as ECC<0.06 and DG>9.9 in the operating bands.

All-optical nano logical gates AND, NOR, OR, and NOT based on plasmonic waveguides with Kerr nonlinear cavity

In this paper, all-optical logical gates were designed and simulated based on Kerr nonlinear circular resonators in order to increase the computational power of an optical central processing unit (CPU) using the two-dimensional finite-difference time-domain (FDTD) method. All the logical gates were designed with a simple and single plasmonic composition in which the switch between the gates is done only by changing the intensity and location of the pumps, signal, and monitor. And this is the novelty of this study, which has been optimized by changing the offset and pulse length of the springs to achieve the maximum transmittance between the OFF and ON states. In addition to the above, a manufacturing tolerance simulation was performed for each gate, which takes into account the probability of error in the experimental production process. Overall, the results showed that the difference in transmission rate for AND, NOR, OR, and NOT was 42%, 95%, 55%, and 92% respectively with an ultrafast time response of up to about 50 fs. The benefits of the present study’s design include the number of gates, high transmission rate, manufacturing tolerance, and a smaller device footprint dimension, all of which make it a good base for the future of photonic computational circuits.

Simultaneous excitations of odd and even order resonances in plasmonic metasurfaces for an orthogonal terahertz probe.

Most of the compelling phenomena pertaining to plasmonic metamaterials revolve around the associated odd and even order resonances. However, excitation of odd and even order modes is polarization sensitive, particularly in the case of well-accepted split-ring resonator based terahertz (THz) plasmonic metasurfaces. Such a drawback limits the practical applications of plasmonic metasurfaces across the electromagnetic spectrum. In this context, we experimentally demonstrate multi-split-ring resonator based THz metasurfaces capable of simultaneously sustaining odd and even order resonances when the polarization of the probe beam is altered through 90°. We believe this work should be beneficial in realizing polarization-independent switches and frequency selective surfaces.

Subwavelength dielectric grating structures with tunable higher order resonance for achromatic augmented reality display.

Color non-uniformities caused by a dispersion effect can seriously affect the image quality for a diffractive waveguide display system. In this work, we propose a subwavelength multilayered dielectric grating structure by a rigorous coupled wave analysis as a novel coupling grating, to the best of our knowledge, for waveguide-based near-eye displays to overcome the "rainbow" effect. Such a grating structure exhibits a tunable high-efficiency resonance in first-order diffraction due to resonant coupling of incident light with the grating structure. A further analysis of the resonant behaviors helps us get a clear understanding of the underlying physics for the mode excitation and resonant coupling process. The first-order resonance with a diffraction efficiency of more than 60% can be achieved with the resonant angle continuously shifted to get a large field of view. The resonant angle, diffraction efficiency, and spectral linewidth can be easily tuned by the geometrical parameters of the grating structure.