VIBRATION OF A COMPLIANT TOWER IN THREE-DIMENSIONS

The three-dimensional motion of an offshore compliant tower using both rigid and flexible beam models is studied in this paper. The tower is modelled as a beam supported by a torsional spring at the base with a point mass at the free end. The torsional spring constant is the same in all directions. When the beam is considered rigid, the two-degree-of-freedom model is employed. The two degrees constitute the two angular degrees of spherical co-ordinates, and the resulting equations are coupled and non-linear. When the beam is considered as elastic, three displacements are obtained as functions of the axial co-ordinate and time; again with coupled and non-linear equations of motion. The free and the forced responses due to deterministic loads are presented. The free responses of the rigid and elastic beams show rotating elliptical paths when viewed from above. The rate at which the path rotates depends on the initial conditions. When a harmonic transverse loading is applied in one direction, the displacement in that direction shows subharmonic resonance of order 1/2 and 1/3 while the displacement in the perpendicular direction is affected minimally. Next, in addition to the harmonic load in one direction, a transverse load is applied in the perpendicular direction. The transverse load varies exponentially with depth but is constant with time. It is found that the transverse load affects the transverse displacements in the perpendicular direction minimally.     

Vibration with “dynamic friction”

The free and harmonically forced vibration of an idealized straight beam oscillating in its fundamental mode with frictional clamps at each end is modelled by using a singular differential equation. Locking and chatter phenomena are analyzed and slipping motions integrated for the free system and the continuously slipping steady state response to harmonic excitation is qualitatively discussed.     

Nonlinear dynamics of two harmonically excited elastic structures with impact interaction

In this paper, non-smooth dynamics of two elastic beams excited by harmonic force with impact interaction is studied through analyses, simulations, and experiments. A two degree-of-freedom vibro-impact model is improved by applying the Galerkin approach and Newton's impact law for the two cantilever beams with impact interaction. Numerical analysis is taken to investigate the vibro-impact motions of cantilever beams excited by harmonic force. The l-adding periodic motions and k=1/1, k=2/2, k=3/4, and k=4/4 type of stable periodic motions of the impacted cantilever beam are presented. Poincaré map is established and the Floquet multipliers of the periodic motions are obtained through semi-analytical method to determine the stability of the motions near the bifurcation point. Through associated experiments, typical bifurcations and chaos of the non-smooth system are examined, which are in good agreement with numerical results.     

Non-linear vibrations of three-layer beams with viscoelastic cores,II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.     

Wave propagation in carbon nanotube intramolecular junctions: finite element calculations

The dispersion of longitudinal and transverse waves in (n,0)–(2n,0) intramolecular junctions (IMJs) are investigated using an atomistic finite element method (FEM). The transient responses of IMJs with different connection types subjected to harmonic incident wave were modelled using three-dimensional elastic beams of carbon bonds and point masses. The linkage between the force-field constants of molecular mechanics and input parameters of beam and mass elements was established through the molecular structural mechanics approach. The wave dispersion simulated by FEM shows good agreement with that of the non-local elastic model in a wide frequency range up to the terahertz region. It is shown that both the microstructure of conical part (connection part) and the coupling of longitudinal vibration and transverse vibration brought by the conicity play important roles in the dispersion of longitudinal and transverse wave in a single-walled IMJ. The amplitude decay of longitudinal wave depended on the distance propagating; the wavelength and the structure in connection part are examined. The results show that the dispersion of the decay of the wave amplitude in IMJ with less pentagon–heptagon defects has a better agreement with analytical results of macroscopic conical shell.     

Vibration of a two-member open frame

The dynamics of a two-member open frame structure undergoing both in- and out-of-plane motion is examined. The frames are modelled using the Euler-Bernoulli beam theory and are further generalized by permitting an arbitrary angle between the beams and the attachment of a payload at the end of the second beam. The equations of motion are derived using Hamilton's principle and the orthogonality conditions are presented. It is shown that the in- and out-of-plane motions can be decoupled by including the axial deformation components in the assumed displacement fields. The natural frequencies of the system and the contribution of each member into the system potential energy are examined via numerical examples.     

Non-linear vibrations of three-layer beams with viscoelastic cores I. Theory

Approximate equations of motion are developed for large amplitude motions of three-layer axially restrained unsymmetrical beams with viscoelastic cores. The external force consists of a constant plus an oscillatory term. The combination of this form of forcing and the large amplitude motions cause the beam to respond at multiples of the forcing frequency. This can lead to difficulties in the complex modulus approach to viscoelasticity. These are overcome here through use of hereditary integrals and their relationships with complex moduli. Theoretical results on the frequency response of clamped, symmetrical beams are compared with earlier experimental work. On the whole, reasonable agreement is found.     

Using degenerate parametric interaction of intense acoustic beams to amplify weak signals

The paper considers the degenerate parametric interaction of an intense acoustic pumping beam and a weak signal beam at the subharmonic. The use of a special emitter system with independent signal emission at the harmonic and subharmonic made it possible to study the features of nonlinear interaction both for different amplitude levels and arbitrary phase relations of the fields at these frequencies. Just as predicted in the theory, the experiment showed that signal amplification at the subharmonic hardly occurs at all. It is shown that the use of odd field harmonics, which are absent for a zero amplitude of the signal wave, makes it possible to substantially increase the efficiency of isolating a weak signal wave. The interaction of beams for large and small acoustic Reynolds numbers of the signal wave is studied.     

Efficiency enhancement of a two-beam free-electron laser using a nonlinearly tapered wiggler

A nonlinear and non-averaged model of a two-beam free-electron laser (FEL) wiggler that is tapered nonlinearly in the absence of slippage is presented. The two beams are assumed to have different energies, and the fundamental resonance of the higher energy beam is at the third harmonic of the lower energy beam. By using Maxwell's equations and the full Lorentz force equation of motion for the electron beams, coupled differential equations are derived and solved numerically by the fourth-order Runge-Kutta method. The amplitude of the wiggler field is assumed to decrease nonlinearly when the saturation of the third harmonic occurs. By simulation, the optimum starting point of the tapering and the slopes for reducing the wiggler amplitude are found. This technique can be applied to substantially improve the efficiency of the two-beam FEL in the XUV and X-ray regions. The effect of tapering on the dynamical stability of the fast electron beam is also studied.     

Efficiency enhancement of a two-beam free-electron laser using a nonlinearly tapered wiggler

A nonlinear and non-averaged model of a two-beam free-electron laser(FEL) wiggler that is tapered nonlinearly in the absence of slippage is presented.The two beams are assumed to have different energies,and the fundamental resonance of the higher energy beam is at the third harmonic of the lower energy beam.By using Maxwell’s equations and the full Lorentz force equation of motion for the electron beams,coupled differential equations are derived and solved numerically by the fourth-order Runge-Kutta method.The amplitude of the wiggler field is assumed to decrease nonlinearly when the saturation of the third harmonic occurs.By simulation,the optimum starting point of the tapering and the slopes for reducing the wiggler amplitude are found.This technique can be applied to substantially improve the efficiency of the two-beam FEL in the XUV and X-ray regions.The effect of tapering on the dynamical stability of the fast electron beam is also studied.     

原子力显微镜高次谐波幅度对样品弹性性质表征的研究

轻敲模式下原子力显微镜微悬臂探针在接近其基态共振频率的外加驱动下振荡, 其末端针尖周期性靠近、远离样品, 产生于针尖与样品非线性相互作用过程中的高次谐波信号包含更多的待测样品表面纳米力学特性等方面的信息. 通过理论分析、计算, 系统地研究了针尖与样品接触时间受样品弹性模量的影响, 以及高次谐波幅度与接触时间的关系, 获得了通过高次谐波幅度区分待测样品表面弹性性质差异的规律. 并在自制的高次谐波成像实验装置上, 得到了与理论预期一致的实验结果. 关键词： 轻敲模式原子力显微镜 接触时间 高次谐波幅度 弹性模量     

基于“泵浦-探测”效应的等离子体通道中三次谐波增强与光谱分析

利用“泵浦-探测”非共线双等离子体通道方法实现三次谐波增强。泵浦光与探测光间的耦合作用能有效改善探测光光丝中的饱和效应,克服光强钳制对谐波强度增加的限制。实验中将两束能量分别为4.4和10.2 mJ、中心波长均为810 nm、脉宽均为60 fs的超短脉冲在空气中非共线聚焦,能各自形成光丝且产生微弱三次谐波。当强光在时域上超前于弱光时,强光会预先成丝形成等离子体通道,对后续弱光产生调制,使探测光产生的三次谐波强度明显增强。实验发现270 nm谐波能量增加的显著区域内频谱宽度出现振荡变化,当两光以27.3 mrad小角度相交于几何焦点前约15 mm,且探测光滞后约55 fs时,获得的能量增长倍率达到近70倍,对应谱宽约为5 nm。     

Non-linear vibration analysis of multilayer beams by incremental finite elements,Part II: Damping and forced vibrations

The inclusion of simple damping of viscous type in the incremental variational equation governing the non-linear motions of multilayer beams is described. Various problems of forced non-linear response of three-layer sandwich beams are studied. Plots of the response curves reveal some complicated and interesting variation of the amplitudes of different harmonic terms, especially in the case of an unsymmetrically layered beam in which a quadratic type of non-linearity is observed.     

Model equation for strongly focused finite-amplitude sound beams

A model equation that describes the propagation of sound beams in a fluid is developed using the oblate spheroidal coordinate system. This spheroidal beam equation (SBE) is a parabolic equation and has a specific application to a theoretical prediction on focused, high-frequency beams from a circular aperture. The aperture angle does not have to be small. The theoretical background is basically along the same analytical lines as the composite method (CM) reported previously [B. Ystad and J. Berntsen, Acustica 82, 698-706 (1996)]. Numerical examples are displayed for the amplitudes of sound pressure along and across the beam axis when sinusoidal waves are radiated from the source with uniform amplitude distribution. The primitive approach to linear field analysis is readily extended to the case where harmonic generation in finite-amplitude sound beams becomes significant due to the inherent nonlinearity of the medium. The theory provides the propagation and beam pattern profiles that differ from the CM solution for each harmonic component.     

Photon-photon elastic scattering in the visible domain

We are investigating the direct detection of photon-photon elastic scattering at optical energies. In a first experiment using two high-intensity pulsed laser beams, we have explored the feasibility of the method, and in particular the rejection of back-ground noise. We obtained an upper limit of the photon-photon elastic scattering cross section at 95% confidence level of 10^?39 cm². This limit can be lowered by twenty orders of magnitude by stimulating the scattering by a third beam, and by using high repetition rate existing lasers.     

A HYBRID FINITE SEGMENT/FINITE ELEMENT MODELLING AND EXPERIMENT ON A FLEXIBLE DEPLOYMENT SYSTEM

This paper focuses on the dynamic responses of a flexible deployment system that has a central rigid body and four articulated flexible beams and undergoes locking impact. A hybrid finite segment/finite element model and an experiment are presented for the deploy-ment system. The flexible beam components in the system are modelled with the finite segments connected by massless beam elements, wherein the finite segments describe the inertia of the large rotation flexible beam and the massless elastic elements describe the elas-ticity of the flexible beam by taking the advantage of small deformation in the relative co-ordinate system. To model the internal impacts in the articulate joints due to clearances, a continuous contact force model of locking joint is also proposed. The governing differential-algebraic equations of the system are established by the Newton-Euler method with Lagrange multipliers and are solved with the method of generalized co-ordinate partitioning. To accelerate the numerical integration, a “longitudinal constraint” is suggested to alleviate the stiff problem of the dynamic equations. In addition, a physical model of the deployment system is constructed. The deployment is released by the compressed springs in the joints. A position measuring system of linear CCD cameras is used to measure the large displacement of the system. Correlations between the mathematical model and the experiments are also presented. Reasonable results are obtained.     

The free vibration of compact rotating radial cantilevers

The free vibration of rotating uniform radial cantilever beams of compact cross section is considered, with account taken of centrifugal coupling between motions in the principal elastic planes. For cases other than those in which the principal elastic axes coincide with the equatorial and meridional planes the centrifugal coupling is shown to modify the vibrational behaviour of the compact beam when compared to that of a beam which is infinitely stiff in one principal plane and can result in a considerable reduction in fundamental natural frequency. Results are presented which show how the natural frequencies and mode shapes of the lower modes vary with spin speed for various root offset and cross-sectional configurations.     

Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load

The vibration of beams on foundations under moving loads has many applications in several fields, such as pavements in highways or rails in railways. However, most of the current studies only consider the energy dissipation mechanism of the foundation through viscous behavior; this assumption is unrealistic for soils. The shear rigidity and radius of gyration of the beam are also usually excluded. Therefore, this study investigates the vibration of an infinite Timoshenko beam resting on a hysteretically damped elastic foundation under a moving load with constant or harmonic amplitude. The governing differential equations of motion are formulated on the basis of the Hamilton principle and Timoshenko beam theory, and are then transformed into two algebraic equations through a double Fourier transform with respect to moving space and time. Beam deflection is obtained by inverse fast Fourier transform. The solution is verified through comparison with the closed-form solution of an Euler-Bernoulli beam on a Winkler foundation. Numerical examples are used to investigate:(a) the effect of the spatial distribution of the load, and(b) the effects of the beam properties on the deflected shape, maximum displacement, critical frequency, and critical velocity. These findings can serve as references for the performance and safety assessment of railway and highway structures.     

Vibration and buckling of a double-beam system under compressive axial loading

On the basis of the Bernoulli–Euler beam theory, the properties of free transverse vibration and buckling of a double-beam system under compressive axial loading are investigated in this paper. It is assumed that the two beams of the system are simply supported and continuously joined by a Winkler elastic layer. Explicit expressions are derived for the natural frequencies and the associated amplitude ratios of the two beams, and the analytical solution of the critical buckling load is obtained. The influences of the compressive axial loading on the responses of the double-beam system are discussed. It is shown that the critical buckling load of the system is related to the axial compression ratio of the two beams and the Winkler elastic layer, and the properties of free transverse vibration of the system greatly depend on the axial compressions.     

Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.