A SEMI-ANALYTICAL APPROACH TO THE NON-LINEAR DYNAMIC RESPONSE PROBLEM OF BEAMS AT LARGE VIBRATION AMPLITUDES, PART II: MULTIMODE APPROACH TO THE STEADY STATE FORCED PERIODIC RESPONSE

The semi-analytical approach to the non-linear dynamic response of beams based on multimode analysis has been presented in Part I of this series of papers (Azrar et al., 1999 Journal of Sound and Vibration224, 183-207 [1]). The mathematical formulation of the problem and single mode analysis have been studied. The objective of this paper is to take advantage of applying this semi-analytical approach to the large amplitude forced vibrations of beams. Various types of excitation forces such as harmonic distributed and concentrated loads are considered. The governing equation of motion is obtained and can be considered as a multi-dimensional form of the Duffing equation. Using the harmonic balance method, the equation of motion is converted into non-linear algebraic form. Techniques of solution based on iterative-incremental procedures are presented. The non-linear frequency and the non-linear modes are determined at large amplitudes of vibration. The basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated. The percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution. Also, the percentage contributions of the higher modes to the bending moment near to the clamps are given, in order to determine accurately the error introduced in the non-linear bending stress estimated by different approximations. Solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency. The non-linear deflection shapes in various regions of the solution, the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes. The numerical results obtained here for the non-linear forced response are compared with those from the linear theory, with available non-linear results, based on various approaches, and with the single mode analysis.     

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.     

Mathematical model of the dynamics of micropolar elastic thin beams. Free and forced vibrations

A method of hypotheses has been developed to construct a mathematical model of micropolar elastic thin beams. The method is based on the asymptotic properties of the solution ofan initial boundary value problem in a thin rectangle within the micropolar theory of elasticity with independent displacement and rotation fields. An applied model of the dynamics of micropolar elastic thin beams was constructed in which transverse shear strains and related strains are taken into account. The constructed dynamics model was used to solve problems of free and forced vibrations of a micropolar beam. Free vibration frequencies and modes, forced vibration amplitudes, and resonance conditions were determined. The obtained numerical calculation results show the specific features of free vibrations of thin beams. Micropolar thin beams have a free vibration frequency which is almost independent of the thin beam size, but depends only on the physical and inertial properties of the micropolar material. It is shown for the micropolar material that the free vibration frequency values of beams can be readily adjusted and hence a large vibration frequency separation can be achieved, which is important for studying resonance.     

Generation of an elliptic Bessel beam

In an elliptic coordinate system the solution for a diffraction-free beam is given by a Mathieu beam, which is mathematically complicated and therefore takes considerable computational time and a large memory space. A technique is reported for the generation of elliptic hollow beams, which are actually scaled versions of higher-order Bessel beams. The analysis is based on the fact that stretching coordinates in the space domain results in a contraction of the coordinates in the frequency domain, along with a change in the overall amplitude of the spectrum. The beam, produced holographically, is structurally stable and retains its shape up to approximately 1 m.     

Three Gaussian beam heterodyne interferometer for surface profiling

A novel three Gaussian beam interferometric technique for profiling optical smooth surfaces is presented. The technique is based on the heterodinization of three Gaussian beams, two of them with the same temporal frequency. The first beam is used as a probe beam after being focused and reflected from the surface under test. The second beam is reflected from a reference surface. The third beam is obtained from the first diffraction order of a Bragg cell and thus, it is shifted in its temporal frequency. The three beams are coherently added at the sensitive plane of a photodetector that integrates the overall intensity of the beams. We show analytically that the electrical signal at the output of the photodetector consists of a temporal carrier whose amplitude is a sinusoidal function of the local topography. We include the measurement of the topography of a sample consisting in a blazed-reflecting grating calibrated by means of an atomic force microscope.     

Optimal design of beams in torsion under periodic loading

The optimal design of beams in torsion under harmonically varying torques is discussed. The analysis covers the cases when the excitation frequency is either less than or greater than the fundamental frequency of the beam. The beams analyzed are in the main assumed to have rectangular cross-section but the theory is easily extended to other section shapes. In each case the problem is stated in variational form with the introduction of constraints through Lagrange multipliers. The mathematical analysis of the various problems presented results in a system of non-linear differential equations with associated boundary conditions. The solutions given for some of the cases provide expressions for the design variable and the response, along the length of the beam, in terms of the forcing frequency and some constants which can be determined for the particular problem. The computed results and data are given in tabular form and some optimum profiles are shown graphically.     

Nonlinear size-dependent behaviour of single-walled carbon nanotubes

This paper examines the nonlinear size-dependent behaviour of single-walled carbon nanotubes (SWCNTs) based on the von-Karman nonlinearity and the nonlocal elasticity theory capable of predicting size effects. To this end, based on Hamilton’s principle in the framework of the nonlocal Euler–Bernoulli beam theory, the equation of motion and associated boundary conditions are derived. Then, with the aid of a high-dimensional Galerkin scheme, the nonlinear partial differential equation of motion of the SWCNT is recast into a reduced-order model. The dynamic response of the system is then investigated for two different types of excitation, namely primary and superharmonic excitations. Eventually, the effect of the slenderness ratio, forcing amplitude, and excitation frequency on the motion characteristics of the system is investigated.     

Airy加速光束的局域空间频率及其在光束设计中的应用

Airy加速光束是近年来备受关注的一种新型无衍射光束. 它所具有的自由加速、无衍射及自恢复特性使其在光学微操纵、非线性光学、 电子加速等诸多领域显示出重要的应用价值. 因此, 如何方便高效地生成加速光束成为近年来的一个热点研究内容. 本文对Airy加速光束复振幅分布的空间振荡特性进行了分析, 建立了利用局域空间频率描述其加速特性的理论. 提出了利用零点坐标计算加速光束局域空间频率的方法, 通过非线性拟合给出了可以精确描述Airy光束局域空间频率的解析公式; 确定了加速光束的局域空间频率函数与加速轨迹之间的定量关系, 给出了由给定加速轨迹计算相应的局域空间频率以及加速光束的纯相位函数的一种简单计算方法. 将上述分析结果用于设计产生具有给定加速轨迹的加速光束所需的相位函数, 成功求出了能够产生圆弧形加速轨迹的新型加速光束的纯相位函数的解析表达式. 基于该相位函数设计的纯相位衍射光学元件的模拟衍射结果证明了上述方法的可行性. 关键词： 衍射光学 加速光束 局域空间频率     

Ce:KNSBN光折变类光纤简并四波混频特性研究

研究了Ce:KNSBN光折变类光纤在泵浦光和信号光夹角分别为小角度和大角度两种情况下简并四波混频的基本特性.当夹角较大时,在光折变类光纤内部形成了两个四波混频作用区域,获得了比小角度情况提高4倍的大相位共轭反射率.给出了Ce:KNSBN光折变类光纤中,在入射夹角分别为小角度和大角度两种情况下,相位共轭光反射率分别随信号光光强、两束泵浦光光强比变化的实验结果,并用理论公式进行了拟合,理论分析和实验结果相符.还研究了Ce:KNSBN光折变类光纤四波混频光栅模式和相位共轭光时间响应特性.光折变类光纤的相位共轭响应时间较快,可为秒量级.     

Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated.     

Wavefront measurements of diode laser beams with large dynamic ranges

We propose and demonstrate a method for the measurement of the wavefronts of high-power diode laser beams with large dynamic ranges. Our wavefront sensor consists of a movable pinhole and a wavefront-slope detector. The measurement results show that the wavefront sensor exhibits a large dynamic range of pi/2 to -pi/2 and a high precision on the measured average wavefront slope. The wavefronts of high-power diode laser beams having large divergence angles at arbitrary locations (including near and far fields) can be reconstructed via the wavefront measurement at a given location. The amplitude and phase distributions of a laser beam could be determined from the measured optical field data using diffraction theory. The experimental measurement of the wavefront of a 980 nm diode laser beam will be presented and discussed.     

On the properties of a ‘finite beam’, reflection-type volume hologram

The properties of a volume-phase hologram of the reflection type are studied in the case when the reference and object waves are beams of finite width. Numerical results are obtained by considering a sufficiently large number of terms in a previously derived infinite series solution. The results reproduce Kogelnik's one-dimensional theory [1] when the thickness of the hologram is small in comparison with the widths of the beams but otherwise the finiteness of the recording beams leads to significant variations in the amplitude of the reconstructed beam.     

Coupled bending and torsional vibration of a beam with in-span and tip attachments

In this paper, the coupled flexural-torsional free and forced vibrations of a beam with tip and/or in-span attachments are studied. First, a mathematical model is established, which consists of a beam with several tip attachments, i.e, a tip mass of non-negligible dimensions, a linear spring grounding the tip mass, and a torsional spring connected at the end of the beam. The modal functions of this model and the orthogonality condition among them are derived. For the purpose of verification the properties of the tip attachments are changed, and the numerical results obtained are compared with those given in the relevant literature. Effects of tip mass and distributed mass in-span on natural frequencies and modes are investigated for two cantilever beams with different cross sections. An application of the orthogonality condition in the case of a beam with tip mass is also presented for a forced vibration example.     

On the damped vibratory response of curved viscoelastic beams

An analysis is given of the damped forced vibratory response of viscoelastic beams with small initial curvature in order to explore the interaction of initial curvature with both extensional and shear deformation, as they provide energy dissipation mechanisms for the damped response. A non-dimensional variational statement is developed which governs the problem of the forced vibratory response of slightly curved Timoshenko beams including in-plane deformation, and appropriate parameters are defined. In particular, initial rise and shear parameters are seen to play important roles in the results.Numerical results are presented for the first three modes for both simple and clamped support conditions. In general, even a slight amount of initial curvature has a significant effect in decreasing the damped response for the first mode, and the actual effect is influenced strongly by the shear parameter. Initial curvature has a smaller influence on the higher modes. An understanding of when extensional or shear energy dissipation is dominant for certain ranges of values of the curvature and shear parameters is obtained by actual plots involving the appropriate form of energy dissipation. In general, extensional energy dissipation tends to dominate in the first mode for both support conditions, although the clamped beam tends to favor shear dissipation over a certain range of parameter values. The dominance of the shear mechanism, however, increases markedly for the higher modes for both support conditions.     

Numerical Simulation of Autoresonant Ion Acceleration

Computational and analytic studies of the Autoresonant Acceleration proposal for collective ion acceleration are presented. Linear theory is reviewed, the electrostatic well depth is estimated nonlinearly, and an electron beam envelope equation is derived and solved. Two-dimensional numerical simulation results are given. Together, these calcualtions demonstrate unneutralized electron beam equilibrium in a diverging magnetic guide field, the behaviour of large amplitude slow cyclotron waves in the beam, and the acceleration of test ions over short distances in the wave troughs. In addition, the computer simulations point up the need for improved understanding of the linear theory of radially inhomogeneous noneutral beams and for methods of suppressing radial modulation at the diode-waveguide interface.     

Circular beams

A very general beam solution of the paraxial wave equation in circular cylindrical coordinates is presented. We call such a field a circular beam (CiB). The complex amplitude of the CiB is described by either the Whittaker functions or the confluent hypergeometric functions and is characterized by three parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integrability are studied in detail. Special cases of the CiB are the standard, elegant, and generalized Laguerre-Gauss beams; Bessel-Gauss beams; hypergeometric beams; hypergeometric-Gaussian beams; fractional-order elegant Laguerre-Gauss beams; quadratic Bessel-Gauss beams; and optical vortex beams.     

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.     

Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements

This paper employs the numerical assembly method (NAM) to determine the “exact” frequency–response amplitudes of a multiple-span beam carrying a number of various concentrated elements and subjected to a harmonic force, and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force. First, the coefficient matrices for the intermediate concentrated elements, pinned support, applied force, left-end support and right-end support of a beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix. The graph of dynamic response amplitudes versus various exciting frequencies gives the frequency–response curve for any point of a multiple-span beam carrying a number of various concentrated elements. For the case of zero harmonic force, the above-mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained.     

Green's functions of the forced vibration of Timoshenko beams with damping effect

This paper is concerned with the dynamic solutions for forced vibrations of Timoshenko beams in a systematical manner. Damping effects on the vibrations of the beam are taken into consideration by introducing two characteristic parameters. Laplace transform method is applied in the present study and corresponding Green's functions are presented explicitly for beams with various boundaries. The present solutions can be readily reduced to those for others classical beam models by setting corresponding parameters to zero or infinite. Numerical calculations are performed to validate the present solutions and the effects of various important physical parameters are investigated.