Shock response of viscoelastically damped sandwich plates

Analysis for the transient response of a simply supported three layer viscoelastically damped sandwich plate, subjected to a half sine shock pulse, has been carried out, with account taken of the transverse inertia effects only. The properties of the viscoelastic core material have been represented by those of a four element viscoelastic model. The influences of the variation of various geometrical and physical parameters of the damped sandwich plate on the shock response are investigated. The decay rate of the transverse vibrations of the plate is evaluated in terms of the logarithmic decrement.     

An analytical formulation of the forced responses of damped rings

An analysis in which an interaction formulation is used for the forced vibratory responses of a ring having a number of mass segments adhered to it by a viscoelastic material is presented. The mass segments are discretely distributed around the circumference of the ring, and the excitation is a concentrated vibratory radial force located on the surface of the ring. The mass segments may not have to be identical, nor do their distributions have to be uniform. The analysis can readily be extended to more complicated damped shell structural systems. The driving point mechanical impedances at a location midway between two mass segments for a given damped system are given as an example. These solutions compare very well with experimental data and theoretical results available. Also presented is a comparison of driving point mechanical impedances for two damped systems with different thicknesses of the viscoelastic layers.     

Forced axisymmetric motion of circular,viscoelastic plates

Williams' method for forced motion of elastic systems is applied to circular, viscoelastic plates where the effects of rotatory inertia, transverse shear and time-dependent boundary conditions are included. The viscoelastic material is assumed to have a constant Poisson's ratio. A particular problem is solved for a symmetrically loaded, completely free plate. The material used is vulcanized rubber where the viscoelastic behavior in shear is used in specifying the material parameters of a three-element solid.     

Vibrations of three layered damped sandwich plate composites

Theoretical and experimental results are presented and discussed for the transverse driving point mechanical impedances, as well as for the transfer impedances, of damped composite plates made up of a thin viscoelastic layer sandwiched between two elastic layers. Analytical results are determined by finite element approximations. Due to the elements used and the system to be modeled, several fundamental assumptions or restrictions usually adopted in analytical investigations are removed. The dependence on frequency and temperature of the dynamic properties of the viscoelastic materials is taken into consideration. A companion experiment was conducted, for comparison purposes, on such damped composite plates suspended in air by lightweight elastic shock cords and driven at the center by an electromechanical vibration shaker. Good correlations between the test data and analytical solutions are obtained over a wide frequency range for two configurations.     

Transverse Vibration of Tapered Single-Walled Carbon Nanotubes Embedded in Viscoelastic Medium

Based on the nonlocal theory, Euler-Bernoulli beam theory and Kelvin viscoelastic foundation model, free transverse vibration is studied for a tapered viscoelastic single-walled carbon nanotube (visco-SWCNT) embedded in a viscoelastic medium. Firstly, the governing equations for vibration analysis are established. And then, we derive the natural frequencies in closed form for SWCNTs with arbitrary boundary conditions by applying transfer function method and perturbation method. Numerical results are also presented to discuss the effects of nonlocal parameter, relaxation time and taper parameter of SWCNTs, and material property parameters of the medium. This study demonstrates that the proposed model is available for vibration analysis of the tapered SWCNTs-viscoelastic medium coupling system.     

The decoupling of damped linear systems in free or forced vibration

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the “classical decoupling problem.” Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.     

对称双弹簧振子受迫、有阻尼横振动的混沌行为

对受周期外力驱动的对称双弹簧振子进行了研究,建立了系统的动力学方程,用线性稳定性分析方法讨论了平衡点附近邻域的稳定性,利用数值计算并结合多种分析方法,求解非线性方程和判断解的性质.通过改变系统参数,画出时域图、相图及分岔图等.计算分析和数值实验发现,这个简单的力学系统存在十分丰富的动力学行为(分岔、混沌).理论分析和数值实验结果一致.     

Analysis of the response of damped cylindrical shells carrying discontinuously constrained beam elements

A simple analysis is presented of the forced vibratory response of a cylindrical shell having a number of axial beams adhered to it by a viscoelastic material layer. The attached beams are identical, closely spaced and distributed around the full circumference of the shell. The excitation is a concentrated vibratory force acting radially at the mid-section on the surface of the shell. The end conditions of the shell and the attached beams are all assumed to be simply supported. The effects of the operational temperature and frequency on the viscoelastic material properties are considered. An experiment was conducted, for comparison, on a damped cylindrical shell suspended in air by lightweight elastic shock cords and driven at the mid-section by an electromechanical vibration shaker. Good correlations between the test data and analytical solutions were obtained over a wide range of frequencies.     

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.     

A finite element method for damped acoustic systems: An application to evaluate the performance of reactive mufflers

The approximate equations governing the forced harmonic motion of a damped acoustic system are set up by using a variational principle. Acoustic finite elements are then used in a computer program to study the transmission loss and insertion loss performance of some expansion chamber mufflers. The manner in which the equations are set up allows a number of input and output nodes, and two-dimensional effects involving the influence of transverse acoustic modes to be taken into account. Although only the simplest of elements and coarse mesh sizes are used the resulting accuracy of the solutions is extremely good; thus the method should be a viable one for studying the performance of more complicated mufflers, having variable cross-sections and internal energy dissipation.     

Stationary response of combined linear dynamic systems to stationary excitation

The stationary response of a broad class of combined linear systems to stationary random excitation is determined by the normal mode method. The systems are characterized by a viscously damped simple beam (or string, membrane, thin plate or shell, etc.) connected at discrete points to a multiplicity of viscously damped linear oscillators and/or masses. The solution of the free vibration problem by way of Green functions and the deterministic forced vibration problem by modal analysis for both proportional and non-proportional damping is reviewed. The orthogonality relation for the natural modes of vibration is used to derive a unique relationship between the cross-spectral density functions of the applied forces and the cross-spectral density functions of the generalized forces. Finally, the response spectral density functions and the mean square responses of the beam and oscillators are derived in closed form, exact for the proportionally damped system and approximate for the non-proportionally damped system.     

Study on vibro-acoustical characteristics of damped composite boxlike shells in water

Based on the structural FEM and the acoustic BEM, a numerical model of coupled elastic layer and viscoelastic layer and outside sound field is established and the vibro-acoustical characteristics of damped composite boxlike shells are studied systematically. It can be concluded that the structural vibration responses and the sound radiation are reduced significantly due to the viscoelastic layer and its effects are dependent on the geometric, physical parameters of the layer and the excitation frequency. It is also shown that compared with the bare elastic shells, the influence of the fluid compressibility on the vibration responses of shells covered with a damping layer is not evident and the effects of the free surface and the rigid plane are weakened.     

Damping properties of layered cylindrical shells,vibrating in axially symmetric modes

The damping of cylindrical shells coated with unconstrained layers of viscoelastic material either on one side of the shell (inside or outside) or on both sides is estimated. The basic equations of motion are derived which describe harmonic forced flexural damped vibrations in axisymmetric modes. For pure sinusoidal modes expressions for the overall loss factors are given. The damping properties of cylindrical shells of finite length, coated on the inside or outside, or on both sides (symmetrically or unsymmetrically) are compared. Classical thin shell theory is used for the analysis. It is shown how two-layered damped shells differ from two-layered damped beams. The extent of damping reduction in shells resulting from the fact that the shell cross-section is closed is discussed.     

On the stability properties of a damped oscillator with a periodically time-varying mass

In this paper the vibrations of a damped, linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be considered. Both the free and forced vibrations of the oscillator will be studied. For the free vibrations the minimal damping rates will be computed, for which the oscillator is always stable. The forced vibrations are partly due to small masses, which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator. Not only solutions of the oscillator equations will be constructed, but also stability properties for the free, and for the forced vibrations will be presented for various parameter values. For the external, harmonic forcing case an interesting resonance condition will be derived.     

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.     

ON THE ROLES OF COMPLEMENTARY AND ADMISSIBLE BOUNDARY CONSTRAINTS IN FOURIER SOLUTIONS TO THE BOUNDARY VALUE PROBLEMS OF COMPLETELY COUPLED r TH ORDER PDES

A heretofore unavailable double Fourier series based approach, for obtaining non-separable solution to a system of completely coupled linear r th order partial differential equations with constant coefficients and subjected to general (completely coupled) boundary conditions, has been presented. The method has been successfully implemented to solve a class of hitherto unsolved boundary-value problems, pertaining to free and forced vibrations of arbitrarily laminated anisotropic doubly curved thin panels of rectangular planform, with arbitrarily prescribed (both symmetric and asymmetric with respect to the panel centerlines) admissible boundary conditions and subjected to general transverse loading.Existing solutions such as those due to Navier or Levy are based on the well-known method of separation of variables. Such solutions represent particular solutions whenever the method of separation of variables work, and when these particular solution functions fortuitously satisfy the boundary conditions. For derivation of the complementary solution, the complementary boundary constraints are introduced through boundary discontinuities of some of the particular solution functions and their partial derivatives. Such discontinuities form sets of measure zero.Various cases of lamination, geometry and dynamic response (forced and free vibrations) of a class of thin anisotropic laminated shells (curved panels) have been shown to follow from the above. Six sets of boundary conditions are used to illustrate the present method for the derivation of complementary solutions. Navier-type solutions whenever available form special cases of the present general solution.     

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.     

FORCED VIBRATIONS OF TRIPLY COUPLED, PERIODICALLY AND ELASTICALLY SUPPORTED, FINITE, OPEN-SECTION CHANNELS

An exact analytical method is presented for the analysis of forced vibrations of uniform, open-section, single- and multi-bay periodic channels. The centre of gravity and the shear centre of the channel cross-sections do not coincide, and hence the flexural vibrations in two mutually perpendicular directions and the torsional vibrations are all coupled. The ends of the channels and the periodic intermediate supports are modelled with springs having finite flexural and torsional stiffnesses. Single-point force excitation has been considered throughout the study, although the developed method is also capable of dealing with multi-point excitation. The channels are assumed to be of Euler-Bernoulli type beams. The study also takes the effects of cross-sectional warping into consideration. The developed method is suitable for structurally damped analysis and in addition to yielding forced vibration characteristics; it also straightforwardly reveals the free vibration properties like the mode shapes.     

绝热气缸-导热活塞系统的振动特性

用计算机模拟的方法研究了绝热气缸中导热活塞的振动特性。在自由振动或受迫振动的情况下，系统在总体上都是和一个普通的阻尼振子相似的。与此同时，本也指明并分析了该热力学系统的一些独特而有趣的性质。     

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.