Circular Frequency Research Articles

Dynamic stiffness matrix for free vibration analysis of flexure hinges based on non-uniform Timoshenko beam

As a typical non-uniform beam with complex varying cross-section and large curvature, free vibration analysis of flexure hinges in the state-of-the-art compliant mechanisms is challenging due to their complicated and even unsolvable governing differential equation. Investigation on the free vibration analysis of non-uniform beams was attractive in the past but still with a few types of specific beams or approximate solutions. The contribution of this paper is to formulate the dynamic stiffness matrix for generic non-uniform beams and applied for the first time to the free vibration analysis of flexure hinges. Firstly, the vibration solution of general non-uniform Timoshenko beam is derived by using a recursive integral in power series of circular frequency but not the traditional way in the well-known method of Frobenious. Then, the dynamic stiffness matrix of non-uniform Timoshenko beam is derived based on the variational principle. At last, the new method is applied for the free vibration analysis of flexure hinges and flexure-based compliant mechanisms. Several case studies demonstrate the advantage of the proposed approach in terms of high accuracy, generality and high efficiency (only one element is needed). The obtained results are useful for dynamic design of flexure hinges and can serve as a benchmark to examine the accuracy of other numerical solutions. Due to the closed form and universality of the proposed method, it is also of benefit for free vibration analysis of other non-uniform beams with all kinds of varying cross-section or functionally graded beams.

Dynamic non‐local theory stress analysis of two equal and collinear mode‐I cracks in the orthotropic medium

AbstractIn this study, dynamic stress analysis of two equal and collinear mode‐I cracks in the orthotropic medium is conducted using the non‐local theory, the generalized Almansi's theorem and the Schmidt method. Fourier transform is applied to solve the problem, and the problem is converted to a pair of dual integral equations. The dynamic non‐local stress fields near the crack tips are obtained. The effects of the distance between the two equal and collinear cracks, the circular frequency of the incident waves the lattice parameter on the stress field near the crack tips are discussed. The present solution exhibits no stress singularity at the crack tips in the orthotropic medium, i.e. the dynamic non‐local stress field near the crack tips is finite. The obtained results play an important role in designing new composite structures in engineering.

Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum

Harmonic axial waves in quasiperiodic-generated structured rods are investigated. The focus is on infinite bars composed of repeated elementary cells designed by adopting generalised Fibonacci substitution rules, some of which represent examples of one-dimensional quasicrystals. Their dispersive features and stop/pass band spectra are computed and analysed by imposing Floquet–Bloch conditions and exploiting the invariance properties of the trace of the relevant transfer matrices. We show that for a family of generalised Fibonacci substitution rules, corresponding to the so-called precious means, an invariant function of the circular frequency, the Kohmoto’s invariant, governs self-similarity and scaling of the stop/pass band layout within defined ranges of frequencies at increasing generation index. Other parts of the spectrum are instead occupied by almost constant ultrawide band gaps. The Kohmoto’s invariant also explains the existence of particular frequencies, named canonical frequencies, associated with closed orbits on the geometrical three-dimensional representation of the invariant. The developed theory represents an important advancement towards the realisation of elastic quasicrystalline metamaterials.

Out-of-plane band structure of a two-dimensional dispersive photonic crystal.

We calculated the out-of-plane band structure of a two-dimensional dispersive photonic crystal (PC). To achieve this goal, the plane wave expansion method was implemented in conjunction with a numerical algorithm, the dispersive photonic crystal iterative method. The PC is an array of circular cross-sectional dispersive MgO Lorentz single-pole rods in a square lattice. The frequency bands are calculated starting at Γ as a function of the oblique component of the wave vector. For the lowest frequencies, it was found that the modes bend drastically to the horizon as the dielectric constant ϵ(ω) is increased to a very positive value. For frequencies above the longitudinal optical phonon circular frequency, where ϵ(ω) has very low positive values, the expected degeneration occurs in the transparency window, and a line of modes behaves close to the line of light.

Pochhammer–Chree waves: polarization of the axially symmetric modes

The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of wave polarization on the free surface due to variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed ( $$c_2$$ ) all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree wave can propagate at $$c_2$$ phase speed.

Collisionless Boltzmann equation approach for the study of stellar discs within barred galaxies

We have studied the kinematics of stellar disc populations within the solar neighbourhood in order to find the imprints of the Galactic bar. We carried out the analysis by developing a numerical resolution of the 2D2V (two-dimensional in the physical space, 2D, and two-dimensional in the velocity motion, 2V) collisionless Boltzmann equation and modelling the stellar motions within the plane of the Galaxy within the solar neighbourhood. We recover similar results to those obtained by other authors using N-body simulations, but we are also able to numerically identify faint structures thanks to the cancelling of the Poisson noise. We find that the ratio of the bar pattern speed to the local circular frequency is in the range ΩB∕Ω = 1.77 to 1.91. If the Galactic bar angle orientation is within the range from 24 to 45 degrees, the bar pattern speed is between 46 and 49 km s−1 kpc−1.

Abnormality of the longitudinal Pochhammer–Chree waves in the vicinity of C2 phase speed

The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed and analytical expressions for displacement fields are obtained. The variation of wave polarization on the free surface due to the variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree waves can propagate at this phase speed.

Experimental demonstration of a 16.9 Gb/s link for coherent OFDM PON robust to frequency offset and timing error

Due to its merits of flexible bandwidth allocation and robustness towards fiber transmission impairments, coherent optical orthogonal frequency division multiplexing (CO-OFDM) technology draws a lot of attention for passive optical networks (PON). However, a CO-OFDM system is vulnerable to frequency offsets between modulated optical signals and optical local oscillators (OLO). This is particularly serious for low cost PONs where low cost lasers are used. Thus, it is of great interest to develop efficient algorithms for frequency synchronization in CO-OFDM systems. Usually frequency synchronization proposed in CO-OFDM systems are done by detecting the phase shift in time domain. In such a way, there is a trade-off between estimation accuracy and range. Considering that the integer frequency offset (IFO) contributes to the major frequency offset, a more efficient method to estimate IFO is of demand. By detecting IFO induced circular channel rotation (CCR), the frequency offset can be directly estimated after fast Fourier transforming (FFT). In this paper, circular acquisition offset frequency and timing synchronization (CAO-FTS) scheme is proposed. A specially-designed frequency domain pseudo noise (PN) sequence is used for CCR detection and timing synchronization. Full-range frequency offset compensation and non-plateau timing synchronization are experimentally demonstrated in presence of fiber dispersion. Based on CAO-FTS, 16.9 Gb/s CO-OFDM signal is successfully delivered over a span of 80-km single mode fiber.

On point source influencing Love-type wave propagation in a functionally graded piezoelectric composite structure: A Green’s function approach

Green’s function plays an important role in solving the problems concerning point action or impulse responsible for wave motions in materials. Prime objective of the this article is to investigate the propagation behaviour of Love-type wave influenced by a point source in a composite structure comprising a functionally graded piezoelectric material layer lying over a functionally graded fibre-reinforced material half-space. Green’s function technique is adopted in order to obtain the dispersion equation, which is further reduced to the classical Love wave equation as a particular case of the problem. The effect of increasing thickness of functionally graded piezoelectric material layer on the circular frequency and wave number is unravelled and depicted graphically. Moreover, influence of heterogeneity, piezoelectricity and dielectric constant associated with functionally graded piezoelectric material layer and effect of heterogeneity parameter and corresponding magnification factor concerned with functional gradedness of functionally graded fibre-reinforced material half-space have been reported through numerical computation and graphical delineation. For sake of computation, numerical data of PZT-5H ceramics for the functionally graded piezoelectric material layer and carbon-fibre epoxy-resin for functionally graded fibre-reinforced material half-space have been considered. Comparative study is performed to elucidate the effect of presence and absence of reinforcement in functionally graded half-space on the phase velocity of Love-type wave propagating in composite structure.

Designing Two-list Group Program Driven Algorithm for Channel Navigation in Internet Protocol Television

The internet protocol television took the advantage of the convergence of internet services to provide seamless interactivity, time shifting, video on demand and pay per view services. However, zapping delay is a critical problem that deters the switching intention of terrestrial subscribers and widespread of internet protocol television services. This article presents the design of an effective two-list group program driven algorithm that minimises the channel seek distance and channel seek time in order to reduce zapping delay in internet protocol television. The algorithm groups similar channel programs in related service categories such as news, movies, sports, music and documentary to make it easier for channels with similarly aired programs to be identified rapidly among several channels that an internet protocol television offers. The simulated results of the proposed algorithm show that the average seek distance is significantly reduced by 58% when benchmarked with the state of the art numerical ordering, frequency circular ordering and frequency interleaved ordering.

Generalized exponential, polynomial and trigonometric theories for vibration and stability analysis of porous FG sandwich beams resting on elastic foundations

Abstract The present article investigates the free vibration and elastic stability behaviour of three-dimensional functionally graded sandwich beams featured by two different types of porosity, with arbitrary boundary conditions and resting on Winkler-Pasternak elastic foundations. The investigation is carried out by using the method of series expansion of displacement components. Various hierarchical refined exponential, polynomial, and trigonometric higher-order beam theories are developed in a generalized manner and are validated and assessed against 3D FEM results. The weak-form of the governing equations (GEs) is derived via Hamilton's Principle. The GEs are then solved by using the Ritz method, whose accuracy is significantly enhanced by orthogonalizing the algebraic Ritz functions by virtue of the Gram-Schmidt process. Convergence and accuracy are comprehensively analysed by testing 86 quasi-3D beam theories. Moreover, the effect of significant parameters such as slenderness ratio, volume fraction index, porosity coefficient, elastic foundation coefficients, FG sandwich beam typology as well as boundary conditions, on the circular frequency parameters and critical buckling loads, is discussed.

Erratum: Transmission performance analysis of circular polarisation frequency selective surface and fast accurate optimisation based on equivalent circuit

Erratum: Transmission performance analysis of circular polarisation frequency selective surface and fast accurate optimisation based on equivalent circuit

A linear-to-circular polarization converter based on I-shaped circular frequency selective surfaces**Project supported by the National Natural Science Foundation of China (Grant Nos. 61471387, 61271250, and 61571460).

In this paper, a linear-to-circular polarization converter using a three-layer frequency selective surface based on I-shaped circular structure resonant is presented and investigated. Numerical simulations exhibit that when the normal y-polarized waves impinge on this device propagating towards +z direction, the two orthogonal components of the transmitted waves have a 90° phase difference as well as the nearly equal amplitudes at the resonant frequency of 7.04 GHz, which means that the left-hand circular polarization is realized in transmission. For validating the proposed design, a prototype which consists of 25 × 25 elements has been designed, manufactured and measured. The measured results are in good agreement with the simulated ones, showing that the polarization conversion transmission is over −3 dB in the frequency range of 5.22−8.08 GHz and the axial ratio is below 3 dB from 5.86 GHz to 7.34 GHz.

Transmission performance analysis of circular polarisation frequency selective surface and fast accurate optimisation based on equivalent circuit

In this study, a promising application of frequency selective surface (FSS) operating in circular polarisation is analysed and designed. Both magnitude and phase consistencies between transverse electric (TE) and transverse magnetic (TM) waves are emphasised, especially transmission phase agreement. Equivalent circuit model (ECM) provides a deep physical insight into phase discrepancy between TE and TM waves. The effects of geometrical parameters of FSS on magnitude and phase consistencies are also investigated by ECM. Additionally, an optimisation process is utilised to obtain desired frequency responses for multi-layer FSS design. This process involves iterative correction and update of coarse circuit model on the basis of a few accurate full wave simulations. Moreover, a dual-layer square slot FSS is efficiently designed using this optimisation method. Finally, a prototype of the optimised dual-layer square slot FSS is fabricated and tested to validate the simulated results.

Non‐local dynamic solution to a Mode‐I crack in an orthotropic medium

AbstractThe paper presents the dynamic solution of a Mode‐I crack in an orthotropic medium by use of the non‐local theory, the generalized Almansi's theorem and the Schmidt method. By using Fourier transform, this problem is converted to a pair of dual integral equations. Different from the classical elasticity solution, the present solution exhibits no stress singularity at the crack tips because of the incorporation of non‐local effects. Numerical examples are provided to show the effects of the crack length, the circular frequency of the incident waves and the lattice parameter on the stress field near the crack tips in an orthotropic medium.

Design of very‐low‐profile circular UHF small antenna using characteristic mode analysis

This study presents a very-low-profile circular ultra-high frequency (UHF) small antenna with a radiating body transformed from a monopole. The proposed radiating body is gradually improved by analysing fundamental modes of the structure with different ground planes via characteristic mode analysis (CMA). CMAs demonstrate operating principles of the radiating body so that a magnetic feeding loop is proposed to excite the radiating body. An equivalent circuit of the radiating body and magnetic feeding loop is then established. The relationships between antenna bandwidth, and eigen value and modal significance are derived through physical insight of the CMA. A prototype of the proposed circular antenna operating at 474 MHz is fabricated and measured. It has a radius of 5.2%λ c (λ c is the operating wavelength), and a profile of 5.2%λ c The CMA of the proposed antenna with testing cable at different lengths is applied to understand the discrepancy between simulations and measurements. The analysis and performance evaluation shows that the proposed antenna can be a strong contender for compact sensors for machine-to-machine and internet of things applications.

Full-order and single-parameter searching analysis of coupled flutter instability for long-span bridges

A full-order and single-parameter searching method (F-S method) for analyzing coupled flutter instability of long-span bridges is proposed based on the full-discretized model of structure. Based on the proper approximation of the circular frequency of complex modes, the characteristic equation of the full-order system is expressed as a complex generalized eigenvalue equation that contains only two variables. The equation is used for flutter analysis by solving the complex generalized eigenvalue problem with an efficient simultaneous iteration method directly. Since its computation is reliable and efficient, the application of the proposed method on the flutter problems of long-span bridges is practical. Moreover, the flutter analysis is performed for Jiangyin Yangtze river suspension bridge with 1385 m main span in the completed stage to illustrate the reliability and effectiveness of the proposed method.

Dynamics of silicon nanobeams with axial motion subjected to transverse and longitudinal loads considering nonlocal and surface effects

A microstructure-dependent dynamic model for silicon nanobeams with axial motion is developed by considering the effects of nonlocal elasticity and surface energy. The nanobeam is considered to subject to both transverse and longitudinal loads arising from nanostructural surface effect and all positive directions of physical quantities are defined clearly prior to modeling so as to clarify the confusions of sign in governing equations of previous work. The nonlocal and surface effects are taken into consideration in the dynamic behaviors of silicon nanobeams with axial motion including circular natural frequency, vibration mode, transverse displacement and critical speed. Various supporting conditions are presented to investigate the circular frequencies by a numerical method and the effects of many variables such as nonlocal nanoscale, axial velocity and external loads on non-dimensional circular frequencies are addressed. It is found that both nonlocal and surface effects play remarkable roles on the dynamics of nanobeams with axial motion and cause the frequencies and critical speed to decrease compared with the classical continuum results. The comparisons of the non-dimensional calculation values by present and previous studies validate the correctness of the present work. Additionally, numerical examples for silicon nanobeams with axial motion are addressed to show the nonlocal and surface effects on circular frequencies intuitively. Results obtained in this paper are helpful for the design and optimization of nanobeam-like microstructures based sensors and oscillators at nanoscale with desired dynamic mechanical properties.

Reconstructing the axial stiffness of a symmetric rod with two elastic supports using single mode

The semi-inverse method was used to solve the inverse problem of the vibration of a symmetric rod with two elastic supports. We discuss when the linear density function of the symmetric rod with two elastic supports is a symmetric polynomial function how to construct the axial stiffness of the symmetric rod with two elastic supports for the same symmetric polynomial function, when the symmetric or anti-symmetric mode and the corresponding circular frequency are known. We discuss the axial stiffness of a symmetric rod with two free ends and two fixed ends that was constructed from the symmetric mode or anti-symmetric mode and the associated linear density function.

Miniaturized Circularly Polarized Square Slot Antenna With Enhanced Axial-Ratio Bandwidth Using an Antipodal Y-strip

A novel single-fed, wideband, circularly polarized slot antenna is proposed and fabricated. Wideband circular polarization is obtained by introducing an antipodal Y-strip to a square slot antenna. The feedline is a U-shaped microstrip line that provides a wide impedance bandwidth. The overall size of the antenna is only 28 × 28 mm 2 (0.3 λ o × 0.3 λ o ). A prototype of the antenna is fabricated and tested. The measured bandwidths for the axial ratio (AR <; 3 dB) and relative impedance (|S 11 | <; -10 dB) are 41.3% (from 4.4 to 6.67 GHz) and 84% (from 3.25 to 8 GHz), respectively, and the antenna has a stable radiation pattern and a gain of greater than 3 dBi over the entire circular polarization frequency band.