Adaptive focusing of a high-intensity beam in an irregular medium

Conclusions These model studies show that the performance in adaptive focusing for high-intensity beams in randomly inhomogeneous media is restricted primarily by nonlinear phenomena. The power density at the object is increased by adaptive control and is only slightly dependent on the fluctuation variance for the modes considered and for the range of nonlinearity parameters.In thermal self-action, large-scale spatial distortions predominate, for which one can use adaptive suppression systems with small numbers of control coordinates. In the case of nonlinear refraction in a moving medium, a considerable improvement is obtained by control on the lower aberrations, i.e., by means of the wavefront tilt and curvatures.A current problem in optimizing adaptive-optics systems is evidently the determination of a sufficient set of control coordinates for focusing a high-intensity beam propagating under real conditions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 30–41, November, 1985.     

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.     

Coupled tapering/uptapering of soliton pairs in nonlinear media

Phenomenon of coupled tapering/uptapring of two mutually incoherent beams coaxially co-propagating in a nonlinear medium with small gain or loss has been investigated in this paper using standard parabolic equation approach (PEA) and the results are compared with the results obtained by Beam Propagation Method (BPM), i.e., by direct simulations of the underlying Nonlinear Schrödinger Equation (NLSE). The PEA results are shown to be in excellent agreement with the BPM results. It is seen that both beams of the pair induce uptapering in each other in presence of losses and tapering in presence of gain. When the medium offers gain to the first beam and losses to the other, both beams taper. When the medium offers gain/absorption to only one of the two beams, the beam undergoes self-tapering/self-uptapering and induces a taperd/uptaperd waveguide. The other beam (for which the medium is lossless) uptapers/tapers due to the taperd/uptaperd waveguide created by the first beam.     

Vibration of bimodular sandwich beams with thick facings: A new theory and experimental results

This study deals with both analytical and experimental investigations of three-layer beams with cores of polyurethane foam and facings of unidirectional cord-rubber. Both of these materials are bimodular (i.e., having different behavior in compression as compared to tension). The new theory presented is a shear-flexible laminate version of the well-known Timoshenko beam theory, which, due to the bending-stretching coupling present in the bimodular case, results in a coupled sixth-order system of differential equations. In this theory, a separate derivation is presented for the shear correction factor. Due to the discontinuities in the normal stress distribution and the bimodularity, the shear correction factor is much different than the classical homogeneous material value of $\text{5}\text{6}$. Theoretical and experimental results are presented for the frequencies of the first three modes of vibration for a pin-ended beam without axial restraint. This work is believed to be the first devoted to vibration of bimodular materials in a sandwich configuration.     

Study of the Surge Signals in a Plasma-Filled Rectangular Cavity

The aim of this analytical study of a plasma-filled rectangular cavity in time domain is to exhibit the ability of the evolutionary approach to study the electromagnetic fields forced by surge signals in a dynamical system. Maxwell’s equations for the fields and the boundary conditions for the perfect electric conductor rectangular cavity are supplemented with the constitutive relation for the plasma. Two different pulse waveforms were used for modeling of the surge signals exciting the fields. The solution is obtained for the dynamical system in the form of product of two elements. First element that depends on coordinates is a modal basis. The other element depending on time is a modal amplitude. The modal basis is specified as a summation of four subspaces. Two of these subspaces resemble the solenoidal modes, and the other two resemble the irrotational modes. Evolutionary differential equations with initial conditions are obtained and solved analytically for the amplitudes.     

Kerr非线性介质中聚焦像散高斯光束的传输特性

当高功率激光通过Kerr非线性介质传输时,Kerr效应会严重影响激光的传输特性.实际应用中常遇到像散光束.迄今为止,像散光束传输特性的研究大都局限于在线性介质中的传输,而在非线性介质中传输的研究较少,且还未涉及像散激光束通过含光学系统的Kerr非线性介质传输变换的研究.本文主要研究Kerr效应对聚焦光束像散特性和焦移特性的影响,以及聚焦像散高斯光束的自聚焦焦距和光束焦点调控.在光束扩展情况下,推导出了聚焦像散高斯光束在Kerr非线性介质中传输的束宽、束腰位置和焦移的解析公式,研究表明:在自聚焦介质中,随着自聚焦作用增强(如光束功率增强),光束像散越强,但焦移越小;在自散焦介质中,随着自散焦作用增强(如光束功率增强),光束像散越弱,但焦移越大.另一方面,在光束自聚焦情况下,推导出了自聚焦焦距的解析公式,研究表明利用光束像散可以调控光束焦点个数.     

Analysis of nonlinear steady state vibration of a multi-degree-of-freedom system using component mode synthesis method

In this paper, an analytical approach for nonlinear forced vibration of a multi-degree-of-freedom system is proposed using the component mode synthesis method. The whole system is divided into some components and a nonlinear modal equation of each component is derived using the free-interface vibration modes. The modal equations of all components and the conjunction conditions are solved simultaneously, and then the modal responses of components are derived. Finally, the dynamic responses of the whole system can be obtained. The degrees of freedom of modal equations can be reduced when the lower vibration modes are only adopted in each component. As a numerical example, a nine-degree-of-freedom system is considered, in which all spring have cubic type nonlinearity. As a result, it is shown that when there are no rigid modes in components, the compliance by the proposed method agrees very well with the exact one even if the lower vibration modes of components are only adopted. The other hand, in the case with rigid modes in components, the compliance has a little error compared with the exact result. It is recognized that the method proposed is very effective in the case without rigid modes in components for the actual application.     

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.     

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.     

Efficient modal control strategies for active control of vibrations

Some efficient strategies for the active control of vibrations of a beam structure using piezoelectric materials are described. The control algorithms have been implemented for a cantilever beam model developed using finite element formulation. The vibration response of the beam to an impulse excitation has been calculated numerically for the uncontrolled and the controlled cases. The essence of the method proposed is that a feedback force in different modes be applied according to the vibration amplitude in the respective modes i.e., modes having lesser vibration may receive lesser feedback. This weighting may be done on the basis of either displacement or energy present in different modes. This method is compared with existing methods of modal space control, namely the independent modal space control (IMSC), and modified independent modal space control (MIMSC). The method is in fact an extension of the modified independent space control with the addition that it proposes to use the sum of weighted multiple modal forces for control. The proposed method results in a simpler feedback, which is easy to implement on a controller. The procedure is illustrated for vibration control of a cantilever beam. The analytical results show that the maximum feedback control voltage required in the proposed method is further reduced as compared to existing methods of IMSC and MIMSC for similar vibration control. The limitations of the proposed method are discussed.     

Propagation properties of vector Mathieu-Gauss beams

The vector Helmholtz-Gauss (vHzG) beam is known as a general family of localized vector beam solutions of the Maxwell equations in the paraxial limit and the vector Mathieu-Gauss beam constitutes its version in elliptic cylindrical coordinates system. In this work, starting from the expansion of the scalar Mathieu-Gauss beam in term of Bessel-Gauss beams, we give a general expression of vector Mathieu-Gauss beams in cylindrical coordinates. Within the frame work of the Collins diffraction integral formula we derive the analytical expressions of transverse vector Mathieu-Gauss beams through an axisymmetric ABCD optical system. Some numerical calculations are performed to illustrate the propagation of the vector Mathieu-Gauss beam in free space and through a simple lens system. The results are analyzed and discussed.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

Generalized nonlinear coupled mode equations for ultrashort pulse propagation in optical fibers

A generalized model to describe the ultrashort pulse propagation in optical fibers with arbitrary modes present by taking advantage of the coupled mode approach is proposed. First the generalized nonlinear coupled mode equations in the frequency domain are derived exactly. Then the simplified forms for the weakly guiding optical fibers are obtained both in the frequency domain and in the time domain. Finally we discuss the particular forms in two special cases, i.e., ideal single-mode fibers and birefringent single-mode fibers.     

Modal interaction activations and nonlinear dynamic response of shallow arch with both ends vertically elastically constrained for two-to-one internal resonance

This study investigates the two-to-one internal resonance of the shallow arch with both ends elastically constraining, and the primary resonance case is considered. The full-basis Galerkin method and the multi-scale method are applied to obtain the modulation equations. It is shown that the natural frequencies of the first two modes cross/avoid to each other when the stiffness of elastic supports at two ends is the same/different. Moreover, the nonlinear modal interactions between these two modes may not/may be activated. The force/frequency-response curves are employed to explore the nonlinear response of the elastically supported shallow arch. The saddle-node bifurcation points and Hopf bifurcation points are observed in these cases. Moreover, the dynamic solutions, i.e., the periodic solution, quasi-periodic solution and chaotic solution are discussed. The numerical simulations are used to illustrate the route to chaos via period-doubling bifurcation.     

An exact frequency equation in closed form for Timoshenko beam clampled at both ends

The author has discovered several errors which are not typographical in the frequency equations for a Timoshenko beam clamped at both ends by Huang who presented the frequency equations and normal mode equations for all six common types of simple, finite beams in closed form for the first time. The exact frequency equations in closed form for Timoshenko beams clamped at both ends are derived based on his analysis. And then in order to justify the amended solutions of Huang, two versions of the closed form exact method and the Ritz method are applied. The frequency equations by the previous researcher present frequencies for only the flexural modes, while the closed form exact method and the Ritz method give ones for the thickness–shear modes as well as the bending modes. The purpose of the present study is to reveal the errors, correct them, and give some numerical results.     

A damping mechanics model and a beam finite element for the free-vibration of laminated composite strips under in-plane loading

A theoretical framework is presented for predicting the nonlinear damping and damped vibration of laminated composite strips due to large in-plane forces. Nonlinear Green-Lagrange axial strains are introduced in the governing equations of a viscoelastic composite and new nonlinear damping and stiffness matrices are formulated including initial stress effects. Building upon the nonlinear laminate mechanics, a damped beam finite element is developed. Finite element stiffness and damping matrices are synthesized and the static equilibrium is predicted using a Newton-Raphson solver. The corresponding linearized damped free-vibration response is predicted and modal frequencies and damping of the in-plane deflected strip are calculated. Numerical results quantify the nonlinear effect of in-plane loads on structural modal damping of various laminated composite strips. The modal loss-factors and natural frequencies of cross-ply Glass/Epoxy beams subject to in-plane loading are measured and correlated with numerical results.     

Self-induced high-speed modulation in microchip solid-state lasers with asymmetric end pumping

We applied laser-diode sheetlike end pumping to a multimode Nd:YVO(4) laser and observed high-speed (>400-MHz) modulation of the intensity of chaotic pulsation near 1 MHz. The frequencies of modulation were the beat frequencies for pairs of closely spaced lasing modes. Asymmetric optical confinement and the resultant modal interference are shown to lead to oval-hollow-mode operation in which modal beat notes induce high-speed modulation, the frequency range of which is 2 orders of magnitude higher than the intrinsic relaxation oscillation frequency. Good numerical reproduction of the observed chaotic pulsations and their high-speed modulation was obtained with model equations in which such effects as nonlinear gain coupling among modes and field interference between pairs of modes were included. High-speed pulsations in nonchaotic lasers were also demonstrated.     

Free vibrations of a laminated beam by a microstructure theory

The free vibrations of a laminated beam are considered within the framework of a theory that models the composite beam as a macrohomogeneous beam with microstructure. The beams are assumed to consist of several parallel alternating layers of two homogeneous, isotropic elastic materials. The system of three coupled partial differential equations is solved exactly, and attention is devoted to the determination of natural frequencies of vibration of laminated beams with (i) hinged-hinged ends and (ii) clamped-clamped ends. For the sake of comparison, the same boundary value problems are also solved within the framework of the so-called effective modulus theory, which treats the composite as a transversely isotropic and “fictitiously” homogeneous Timoshenko beam, with effective moduli and density. For relatively long beams, i.e., in the low frequency range, the natural frequencies obtained from the two theories are in excellent agreement, but as the depth-to-length ratio, ζ, increases the microstructure frequencies are observed to be much lower than the effective modulus frequencies, the magnitude of the effect becoming more pronounced with increasing mode number n.