From the inverted pendulum to the periodic interface modes

The physics of the spatial propagation of monochromatic waves in periodic media is related to the temporal evolution of the parametric oscillators. We transpose the possibility that a parametric pendulum oscillates in the vicinity of its unstable equilibrium position to the case of monochromatic waves in a lossless unidimensional periodic medium. We develop this concept, that can formally applies to any kind of waves, to the case of longitudinal elastic wave. Our analysis yields us to study the propagation of monochromatic waves in a periodic structure involving two main periods. We evidence a class of phonons we refer to as periodic interface modes that propagate in these structures. These modes are similar to the optical Tamm states exhibited in photonic crystals. Our analysis is based on both a formal and an analytical approach. The application of the concept to the case of phonons in an experimentally realizable structure is given. We finally show how to control the frequencies of these phonons from the engineering of the periodic structure.     

On the harmonic oscillator with nonzero periodic mass

Summary Following previous papers of Leach, and Wille and Vennik, we consider various new expressions for the nonvanishing periodic mass parameter of the time-dependent harmonic oscillator in a Pérot-Fabry cavity in contact with an atomic reservior. The authors of this paper have agreed to not receive the proofs for correction     

Vibration of high-speed rotating rings coupled to space-fixed stiffnesses

This study investigates the vibration of high-speed rotating rings coupled to space-fixed discrete stiffnesses. The ring radial and tangential deformations are defined using space-fixed (Eulerian) coordinates, where material particles pass through fixed locations in space. Engineering strain is used in the strain energy expression. The derived nonlinear equations from Hamilton's principle are linearized about the steady non-trivial configuration that results from constant ring rotation. Comparisons are made to other models in the literature that use different assumptions. The governing equations are cast in terms of matrix differential operators that reveal the system's standard gyroscopic system structure. The natural frequencies and vibration modes are calculated over a wide-range of rotation speeds for axisymmetric free rings and a non-axisymmetric ring with a space-fixed discrete stiffness element.     

On the eigenvalues of the rotating harmonic oscillator

The eigenvalues of the radial Schrödinger equation of a rotating harmonic oscillator depending on a parameter α > 0 are shown to be independent of α and equal to integers.     

On the black hole limit of rotating discs and rings

Solutions to Einstein’s field equations describing rotating fluid bodies in equilibrium permit parametric (i.e. quasi-stationary) transitions to the extreme Kerr solution (outside the horizon). This has been shown analytically for discs of dust and numerically for ring solutions with various equations of state. From the exterior point of view, this transition can be interpreted as a (quasi) black hole limit. All gravitational multipole moments assume precisely the values of an extremal Kerr black hole in the limit. In the present paper, the way in which the black hole limit is approached is investigated in more detail by means of a parametric Taylor series expansion of the exact solution describing a rigidly rotating disc of dust. Combined with numerical calculations for ring solutions our results indicate an interesting universal behaviour of the multipole moments near the black hole limit.     

Contribution to harmonic balance calculations of self-sustained periodic oscillations with focus on single-reed instruments

The harmonic balance method (HBM) was originally developed for finding periodic solutions of electronical and mechanical systems under a periodic force, but has been adapted to self-sustained musical instruments. Unlike time-domain methods, this frequency-domain method does not capture transients and so is not adapted for sound synthesis. However, its independence of time makes it very useful for studying any periodic solution, whether stable or unstable, without care of particular initial conditions in time. A computer program for solving general problems involving nonlinearly coupled exciter and resonator, HARMBAL, has been developed based on the HBM. The method as well as convergence improvements and continuation facilities are thoroughly presented and discussed in the present paper. Applications of the method are demonstrated, especially on problems with severe difficulties of convergence: the Helmholtz motion (square signals) of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring.     

带开口磁环的周期永磁聚焦系统的2维模拟

由于目前常用的磁聚焦系统设计方法不能直接计算带有非轴对称开口磁环的磁系统,因此在行波管磁系统实验测试基础上,提出了双面带极靴的开口磁环体积等效概念：将开口磁环等效为相等体积磁性材料的2维轴对称磁环,两者都双面带极靴时轴上的磁场值相等。使用2维电磁计算软件FEMM计算了采用该等效模型后的周期磁聚焦系统的轴上磁场值,计算结果与带有开口磁环的实际系统测试值符合良好,两者相差仅3×10-3 T。该2维等效模型能代替开口磁环进行计算。     

带开口磁环的周期永磁聚焦系统的2维模拟

 由于目前常用的磁聚焦系统设计方法不能直接计算带有非轴对称开口磁环的磁系统,因此在行波管磁系统实验测试基础上,提出了双面带极靴的开口磁环体积等效概念：将开口磁环等效为相等体积磁性材料的2维轴对称磁环,两者都双面带极靴时轴上的磁场值相等。使用2维电磁计算软件FEMM计算了采用该等效模型后的周期磁聚焦系统的轴上磁场值,计算结果与带有开口磁环的实际系统测试值符合良好,两者相差仅3×10^-3 T。该2维等效模型能代替开口磁环进行计算。     

Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for the relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Making a change of variable, a new nonlinear differential equation is obtained and two procedures are used to approximately solve this differential equation. In the first the differential equation is rewritten in a form that does not contain a square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the first-order approximate frequency obtained by means of the second procedure and the exact frequency never exceeds 1.6%. We also obtained the approximate frequency by applying the second-order harmonic balance method and in this case the relative error is as low 0.31% for all the range of values of amplitude of oscillation A.     

Response of high-purity titanium to high-pressure impulsive loading

Abstract

Measurements of free surface velocity profiles of high-purity titanium samples under shock-wave loading were performed to study the dynamic strength and phase transition parameters. The peak pressure of the initial compression waves was within the range of 4 to 40 GPa, and the load duration was vaned between 10^?8 and 10^?6 s. An anomalous structure of shock waves was observed at pressures of ～ 2.0 to 5.0 GPa due to the α-ω phase transition. The dynamic strength of pure titanium is lower than that of titanium alloys but exceeds the spall strength of commercial grade titanium.     

外周期力驱动的倒摆混沌运动演示仪

设计并制作了受周期外力驱动的倒摆演示实验装置．利用自主开发的软件可自动演示和记录摆球运动状态的时间序列并存储实验数据．通过改变摆长、驱动电压，观察到该系统存在通过倍周期分岔通向混沌的过程．     

Mono-coupled periodic systems with a single disorder: Response to convected loadings

This paper presents a general theory of the forced response under convected loading of mono-coupled periodic systems with a single disorder. The general expressions derived have been used to study the response of an infinite periodic beam on simple supports with one of the support spacings different from all the others. Convected harmonic pressure fields and frozen random pressure fields have been considered. Computer studies are presented showing the moment response at supports and the space-time-averaged responses in the disorder and in the nearby periodic beam elements. High response levels can occur due to (i) resonances of the beam length disorder against the stiffness of the attached periodic systems and (ii) hydrodynamic coincidence vibration occurring in the periodic beam. The frequency zones in which these high responses may occur are identified. The high response due to the resonance (ii) is restricted to the vicinity of the disorder, whereas that due to coincidence occurs throughout the system. Computed results show that the highest response levels do not necessarily occur in the beam length disorder, but may occur in one of the nearby periodic beam elements. The dependence of the maximum response levels on the magnitude of the disorder has been investigated. The conditions under which small disorders may be neglected have been pointed out.     

Comments on a paper concerning the eigenvalues of the rotating harmonic oscillator

It is pointed out that a paper by Flessas on the eigenvalues of the rotating harmonic oscillator recently published in Physics Letters, contains wrong conclusions.     

Structure and magnetic field of periodic permanent magnetic focusing system with open magnetic rings

The magnetic field along the central axis for an axially magnetized permanent magnetic ring was investigated by analytical and finite element methods. For open magnetic rings, both calculated and measured results show that the existence of the radial magnetic field creates a remarkable cosine distribution field along the central axis. A new structure of periodic permanent magnet focusing system with open magnetic rings is proposed. The structure provides a satisfactory magnetic field with a stable peak value of 120 mT for a traveling wave tube system.     

Dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading

The dynamic instability characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element analysis. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The method of Hill's infinite determinants is applied to determine the dynamic instability regions. Numerical results are presented to study the effects of various parameters, such as static load factor, aspect ratio, boundary conditions, stiffening scheme and load parameters on the principal instability regions of stiffened plates using Bolotin's method. The results show that location, size and number of stiffeners have a significant effect on the location of the boundaries of the principal instability region.     

Silicon micro-hemispheres with periodic nanoscale rings produced by the laser ablation of single crystalline silicon

We describe the fabrication of silicon micro-hemispheres by adopting the conventional laser ablation of single crystalline silicon in the vacuum condition without using any catalysts or additives. The highly oriented structures of silicon micro-hemispheres exhibit many periodic nanoscale rings along their outer surfaces. We consider that the self-organized growth of silicon micro-structures is highly dependent on the laser intensity and background air medium. The difference between these surface modifications is attributed to the amount of laser energy deposited in the silicon material and the consequent cooling velocity.