Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods

This paper deals with geometrically nonlinear vibrations of sandwich beams with viscoelastic materials. For this purpose, a new finite element formulation has been developed, in which a zig-zag model is used to describe the displacement field. The viscoelastic behaviour is handled by using hereditary integrals and their relationships with complex moduli. An efficient solution procedure based on the harmonic balance method is also developed. To demonstrate its abilities, various problems of nonlinear vibrations of sandwich beams are considered. First, the results derived from the proposed approach are compared with those of nonlinear dynamic analyses using direct time integration and to experimental data. Then, the influence of the vibration amplitude on the damping properties of sandwich beams is investigated. The effect of an initial axial strain is also examined.     

A finite element analysis of the harmonic response of damped three-layer plates

Solutions have been obtained for the vibration response under harmonic excitation of three-layer plates with a constrained viscoelastic layer (e.g., plates with two metallic outer layers and a viscoelastic core) by means of a finite element method. Damping has been introduced by replacing the real modulus of the viscoelastic material by a complex equivalent which accounts for the phase difference between strain and stress. Triangular finite elements were used with different numbers of degrees of freedom and the dynamic stiffness of the overall structure was calculated. The present method allows for the nonlinear stress-strain behaviour of the viscoelastic material, the effects of the rotatory inertia and the extension within the viscoelastic core. In addition, the use of triangular elements allows for a great variety of shapes and boundary conditions. The finite element computation has been verified by comparison with experimental results for circular three-layer plates and for sandwich beams.     

Micro-vibration suppression of equipment supported on a floor incorporating magneto-rheological elastomer core

In this study, magneto-rheological elastomers (MREs) are adopted to construct a smart sandwich beam for micro-vibration control of equipment. The micro-vibration response of a smart sandwich beam with MRE core which supports mass-concentrated equipment under stochastic support-motion excitations is investigated to evaluate the vibration suppression capability. The dynamic behavior of MREs as a smart viscoelastic material is characterized by a complex modulus dependent on vibration frequency and controllable by external magnetic fields. A frequency-domain solution method for the stochastic micro-vibration response of the smart sandwich beam supporting mass-concentrated equipment is developed based on the Galerkin method and random vibration theory. First, the displacements of the beam are expanded as series of spatial harmonic functions and the Galerkin method is applied to convert the partial differential equations of motion into ordinary differential equations. With these equations, the frequency-response function matrix of the beam–mass system and the expression of the velocity response spectrum are then obtained, with which the root-mean-square (rms) velocity response in terms of the one-third octave frequency band can be calculated. Finally, the optimization problem of the complex modulus of the MRE core is defined by minimizing the velocity response spectrum and the rms velocity response of the sandwich beam, through altering the applied magnetic fields. Numerical results are given to illustrate the influence of MRE parameters on the rms velocity response and the response reduction capacity of the smart sandwich beam. The proposed method is also applicable to response analysis of a sandwich beam with arbitrary core characterized by a complex shear modulus and subject to arbitrary stochastic excitations described by a power spectral density function, and is valid for a wide frequency range.     

PREDICTION AND MEASUREMENT OF SOME DYNAMIC PROPERTIES OF SANDWICH STRUCTURES WITH HONEYCOMB AND FOAM CORES

Some dynamical properties of sandwich beams and plates are discussed. The types of elements investigated are three-layered structures with lightweight honeycomb or foam cores with thin laminates bonded to each side of the core. A six order differential equation governing the apparent bending of sandwich beams is derived using Hamilton's principle. Bending, shear and rotation are considered. Boundary conditions for free, clamped and simply supported beams are formulated. The apparent bending stiffness of sandwich beams is found to depend on the frequency and the boundary conditions for the structure. Simple measurements on sandwich beams are used to determine the bending stiffness of the entire structure and at the same time the bending stiffness of the laminates as well as the shear stiffness of the core. A method for the prediction of eigenfrequencies and modes of vibration are presented. Eigenfrequencies for rectangular and orthotropic sandwich plates are calculated using the Rayleigh-Ritz technique assuming frequency dependent material parameters. Predicted and measured results are compared.     

PARAMETRIC INSTABILITY OF MULTI-LAYERED SANDWICH BEAMS

The present work deals with the parametric instability of multi-layered symmetric sandwich beams with alternate elastic and viscoelastic layers, subjected to periodic axial load. The equations of motion and boundary conditions are derived by applying Hamilton's principle, and the general Galerkin method is utilized to convert the equations of motion into a set of coupled Hill's equations with complex coefficients in the time domain. Numerical results are obtained for beams having, three, five and seven layers. The effects of the shear and core-thickness parameters as well as the core loss factor upon the regions of parametric instability are considered. Zones of instability are obtained for the cases in which beams with various number of layers have the same weight or size or flexural rigidity. Finally, for the criteria of constant size and flexural rigidity as well as constant weight and flexural rigidity, the effects of various parameters on the stability of the system are studied.     

Forced response of FRP sandwich panels with viscoelastic materials

In this paper a new analytical model is presented that accurately predicts the forced response of fibre reinforced plastic (FRP) sandwich plates subjected to transverse applied loads. It is based on Reddy's refined high order shear deformation theory and offers the feasibility of accounting for the viscoelastic properties of the constitutive materials without restriction to the steady state motion. This is achieved by modelling the viscoelastic behaviour of the constitutive materials using the Golla Hughes McTavish mathematical tool. Validation of the new approach is achieved by comparing results under harmonic loading conditions against data obtained using the proposed new analytical model. Subsequently, predicted responses for a given FRP sandwich plate under various transverse applied loads are presented. The results outline the importance of being able to account for the viscoelastic properties of the constitutive materials when modelling the dynamic behaviour of sandwich structures.     

An integral equation analysis of the harmonic response of three-layer beams

The integral equations of harmonic motion have been derived and solved for three-layer sandwich beams with a constrained linear viscoelastic core. The method of solution required first the construction of the Green's vector for a beam in analytical form. Following this, the integral equations were derived and readily approximated by matrix equations which were finally solved numerically. In addition to this analysis, the corresponding eigenvalue problem has been solved so that the modal frequencies and the beam loss factor could be calculated directly. The integral equation analysis offers a fast and efficient alternative to the traditional methods based on the solution of the differential equations of motion. The method has been verified by comparison with experimental results for three-layer cantilevers and simply supported beams.     

The geometric phase in nonlinear frequency conversion

The geometric phase of light has been demonstrated in various platforms of the linear optical regime, raising interest both for fundamental science as well as applications, such as flat optical elements. Recently, the concept of geometric phases has been extended to nonlinear optics, following advances in engineering both bulk nonlinear photonic crystals and nonlinear metasurfaces. These new technologies offer a great promise of applications for nonlinear manipulation of light. In this review, we cover the recent theoretical and experimental advances in the field of geometric phases accompanying nonlinear frequency conversion. We first consider the case of bulk nonlinear photonic crystals, in which the interaction between propagating waves is quasi-phase-matched, with an engineerable geometric phase accumulated by the light. Nonlinear photonic crystals can offer efficient and robust frequency conversion in both the linearized and fully-nonlinear regimes of interaction, and allow for several applications including adiabatic mode conversion, electromagnetic nonreciprocity and novel topological effects for light. We then cover the rapidly-growing field of nonlinear Pancharatnam-Berry metasurfaces, which allow the simultaneous nonlinear generation and shaping of light by using ultrathin optical elements with subwavelength phase and amplitude resolution. We discuss the macroscopic selection rules that depend on the rotational symmetry of the constituent meta-atoms, the order of the harmonic generations, and the change in circular polarization. Continuous geometric phase gradients allow the steering of light beams and shaping of their spatial modes. More complex designs perform nonlinear imaging and multiplex nonlinear holograms, where the functionality is varied according to the generated harmonic order and polarization. Recent advancements in the fabrication of three dimensional nonlinear photonic crystals, as well as the pursuit of quantum light sources based on nonlinear metasurfaces, offer exciting new possibilities for novel nonlinear optical applications based on geometric phases.     

More on finite element modeling of damped composite systems

Finite element procedures are developed and verified for layered beams and rings having either continuously or discontinuously constrained viscoelastic damping layers. The two configurations considered are (1) a three-layered sandwich beam or ring (closed curved beam) consisting of two thin elastic layers with a viscoelastic core in between, and (2) a damped composite made of a thin-walled elastic structure having a finite number of mass segments or elastic segments adhered to it by a viscoelastic material. Viscoelastic material dependence on frequency and temperature is accounted for. Numerical predictions of transverse driving point impedances agree very well with available experimental data.     

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.     

二次谐波产生的非线性相移的解析研究

给出一种分析二阶非线性级联效应的解析方法，对Ⅱ型二次谐波产生的非线性级联效应进行了详细的理论研究和分析。结果表明倍频晶体中正交输入两个振幅不相等的基波分量时，在近相位匹配下二阶非线性级联效应能够产生大的相移。利用本文的解析方法可以对任意二阶非线性相移产生的过程进行优化，从而以最小的需求输入功率来获得特定的相移。还对由非线性相移诱导的且与光强有关的偏振旋转效应和一种基于偏振旋转效应的全光偏振开关进行了分析和讨论。本文的结果对于建立一种基于Ⅱ型倍频晶体中非线性相移诱导偏振旋转效应的新全固化被动锁模飞秒激光装置具有十分重要的意义。     

Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity

The nonlinear oscillations of spherical gas bubbles in linear viscoelastic fluids are studied. A novel approach is implemented to derive a governing system of nonlinear ordinary differential equations. The linear Maxwell and Jeffreys models are chosen as the fluid constitutive equations. An advantage of this new formulation is that, when compared with previous approaches, it facilitates perturbation methods and numerical investigations. Analytical solutions are obtained using a multiple scale perturbation method and compared with the Newtonian results for various Deborah numbers. Numerical analysis of the full equations supports the perturbation analysis, and further reveals significant differences between the viscoelastic and Newtonian cases. Differences in the oscillation phase and harmonic structure characterize some of the viscoelastic effects. Subharmonic excitations at particular fluid parameters lead to a discrete group modulation of the radial excursions; this appears to be a unique, previously undiscovered phenomenon. Implications for medical ultrasound applications are discussed in light of these current findings.     

Rayleigh-Lamb wave propagation on a fractional order viscoelastic plate

A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed.     

Generation of the Second Harmonic by Bessel Light Beams Under the Conditions of Quasisynchronism

The specific features of the doubling of Bessel light beams in quasisynchronous interaction in periodically polarized nonlinear crystals are investigated. A theoretical model has been developed, which is based on the representation of the second-harmonic field in the form of superposition of Bessel beams. Based on the analysis of overlapping integrals, two groups of modes responsible for the formation of the central and circular beams of the second harmonic under the conditions of quasisynchronism were distinguished. It is found that, unlike homogeneous media, crystals with periodic polarization can be used for doubling the frequency of Bessel beams with cone angles changing in wide limits.     

The equation of state of a microinhomogeneous medium and the frequency dependence of its elastic nonlinearity

In the framework of a rheological model, a nonlinear dynamic equation of state of a microinhomogeneous medium containing nonlinear viscoelastic inclusions is derived. The frequency dependences of the effective nonlinear parameters are determined for the difference frequency and second harmonic generation processes in the case of a quadratic elastic nonlinearity. It is shown that the frequency dependence of the nonlinear elasticity of the medium is governed by the linear relaxation response of the inclusions at the primary excitation frequency, as well as by the relaxation of the inclusions at the nonlinear generation frequencies.     

Structure of a nonparaxial gaussian beam near the focus: III. Stability,eigenmodes, and vortices

Exact analytical structurally stable solutions of the Maxwell equations for singular mode beams propagating in free space or a uniform isotropic medium are obtained. Approximate boundary conditions are chosen in the form of the requirement that in the paraxial approximation the fields of nonparaxial mode beams in the waist plane are transformed into the fields of eigenmodes and vortices of weakly guiding optical fibers with the axial symmetry of refractive index. It is shown that optical vortices, in spite of a rather complex structure of field distribution, do not experience substantial changes in the beam form and reproduce, in general features, the field of paraxial vortices. Linear perturbations of the characteristic parameters of mode beams do not change the structure of their electromagnetic field. Nonparaxial singular beams have one more important property, in addition to the fact that the structure of these beams in the paraxial approximation is similar to the structure of the fields of eigenmodes in a fiber. The propagation constants of eigenmodes of a fiber exactly coincide (in the first approximation of perturbation theory) with the projection of the wave vector of a mode beam on the optical axis (an analog of the propagation constant). The possibility of the paraxial transition for nonparaxial mode beams with arbitrary values of azimuthal and radial indices is shown. The properties of nonparaxial modes are illustrated by numerous examples. The solutions obtained and the results of their analysis can be used for exact matching optical fibers and laser beams in various applications.     

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.     

Photoinduced Second Harmonic Generation of CuI Microcrystalite Doped Glass

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Frequency dependence of quantum path interference in non-collinear high-order harmonic generation

High-order harmonic generation(HHG) driven by two non-collinear beams including a fundamental and its weak second harmonic is numerically studied. The interference of harmonics from adjacent electron quantum paths is found to be dependent on the relative delay of the driving pulse, and the dependences are different for different harmonic orders.This frequency dependence of the interference is attributed to the spatial frequency chirp in the HHG beam resulting from the harmonic dipole phase, which in turn provides a potential way to gain an insight into the generation of high-order harmonics. As an example, the intensity dependent dipole phase coefficient α is retrieved from the interference fringe.     

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller–Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity suppresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications.