Vibration of an excitation system supported flexibly on a viscoelastic sandwich beam at its mid-point

A vibration analysis of an excitation system supported flexibly on a three layer sandwich beam is presented in this paper. The flexibly supported excitation system, which is essentially the primary system, consists of a mass, a spring and a dash-pot. The beam is analyzed separately as a continuous system in a classical way and then its dynamic stiffness at the junction point is combined with that of the primary system to obtain the resultant dynamic stiffness, which in turn is used to develop the expressions for the response of the primary system and the transmissibility provided by the whole system. Both response and transmissibility are evaluated for different geometrical and physical parameters of the sandwich beam. The solution to this problem is also obtained by approximating the sandwich beam by a lumped mass supported on a spring and dash-pot. The results in the two cases are compared. Results obtained from an experimental test-rig substantiate the theoretical results.     

轴向加速运动黏弹性梁受迫振动中的混沌动力学

研究了黏弹性轴向运动梁在外部激励和参数激励共同作用下横向振动的混沌非线性动力学行为. 引入有限支撑刚度, 并考虑黏弹性本构关系取物质导数, 同时计入由梁轴向加速度引起的沿径向变化的轴力, 建立轴向运动黏弹性梁横向非线性振动的偏微分-积分模型. 通过Galerkin截断方法研究了外部激励的频率和因速度简谐脉动引起的参数激励的频率在不可通约关系时轴向运动连续体的非线性动力学行为, 并对不同截断阶数的数值预测进行了对比. 基于对控制方程的Galerkin截断, 得到离散化的常微分方程组, 使用四阶Runge-Kutta方法求解. 基于此数值解, 运用非线性动力学时间序列分析方法, 通过Poincaré 映射, 观察到轴向运动梁随扰动速度幅值的倍周期分岔现象, 并比较了有无外部激励对倍周期分岔的影响. 分别在低速以及近临界高速运动状态下, 从相平面图、Poincaré 映射以及频谱分析的角度识别了系统中存在的准周期运动形态. 关键词： 轴向运动梁 非线性 混沌 分岔     

Measurement of dynamic properties of viscoelastic materials

A new method for measurement of viscoelastic properties of materials, based on the exact solution of the problem of the forced oscillations of a flat sample loaded by inertial mass, is proposed. The device for these measurements is described. The elasticity modulus and loss tangent are measured within the frequency range from 100 Hz to 10 kHz at relative deformation of a sample below 1%. The approximate formulas for calculation of viscoelastic properties are derived, and the conditions of their applicability are determined. The methods for expansion of the frequency range and measurement of materials with extremely high loss tangent are proposed. The proposed method is compared with the current Standard. It is shown that the new method has the higher accuracy and expanded range of parameter measurements, and the construction of device is easy and reliable, since it does not require the knowledge on the oscillation properties of vibration source. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 06-08-00193-a).     

Non-linear free vibration of a beam with time-dependent material properties

An account is given of a study of free vibrations of a simply supported beam with a central mass and undergoing creep deformation for the two cases (i) creep deformation is completely recoverable (without hysteresis) and (ii) creep deformation is partly recoverable (with hysteresis). The material is of the Ramberg-Osgood type. Numerical techniques were used to find the response of the system. The results indicate that the beam behaves like a soft spring and that amplitude and frequency decrease with time.     

Acoustic radiation from a viscoelastic beam impacted by a steel sphere

The acoustic radiation from a viscoelastic beam impacted by a steel sphere has been studied both theoretically and experimentally. Transverse vibrations of free-free viscoelastic beams have been analyzed by employing the modal analysis technique and an approximate method, with the Hertz theory used to evaluate impact forces. The wave equation was solved to determine the acoustic pressure radiated from impacted beams of circular and elliptical cross-sections. The theoretical predictions are compared with the experimental results for the radiation from PMMA beams of circular and rectangular cross-sections. It is shown that for beams of circular cross-sections the theoretical and experimental results are in good agreement and that for beams of rectangular cross-sections the radiation is well predicted by modeling them as beams with elliptical cross-sections.     

Dynamic torsion of a two-layer viscoelastic beam

The complex dynamic shear modulus of soft polymeric materials may be determined in principle at low and audio frequencies from torsion pendulum and torsional resonance experiments on metal strips coated with the polymer. In order to determine the polymer shear properties from such experiments, it is necessary to know the torsional rigidity of the two-layer compound beam. This is calculated in the paper by using classical elasticity theory, for the particular case when the metal shear modulus is much greater than that of the soft coating. The theory of dynamic torsion tests is then briefly reviewed and experiments are suggested for determining the polymer dynamic shear properties. A discussion is also given of the effectiveness of polymer coatings as a damping treatment for torsional vibrations.     

Validation of estimated isotropic viscoelastic material properties and vibration response prediction

This paper is concerned with finite element (FE) prediction of forced vibrations using a linear viscoelastic constitutive vibration damping modelling technique. A combined numerical and experimental investigation was performed on two bonded aluminium-PMMA (polymethyl methacrylate) plates with different geometry. Three-dimensional FE models were established using experimentally estimated PMMA material properties (elastic and damping) from previously published procedures. The viscoelastic material damping parameters are here validated from the perspective of accurate estimation of constitutive material properties. Vibration responses were predicted from the FE models and measured on the two composite plate structures at a large number of points. Comparisons between the numerical FE simulations and corresponding measured responses show that the estimated material damping properties used as input to the computations are very accurate and may be treated as independent of the geometry and boundary conditions of the plate structures, i.e., as constitutive damping parameters.     

Residual vibration reduction of a flexible beam deploying from a translating hub

This paper presents a method for reducing the residual vibration of a flexible beam deployed from a translating hub. Whereas previous studies have discussed reducing vibration in translating constant-length beams, this study investigates a vibration reduction method for translating beams of variable length. The partial differential equation of motion for a translating beam is derived and transformed into a variational equation. Based on the discretized equations from the variational equation, the dynamic responses of the flexible beam under translation are analyzed. A vibration reduction method is proposed that is effective for both constant- and variable-length deploying translating beams.     

Frictional vibration transmission from a laterally moving surface to a traveling beam

As the density of information stored in automated magnetic tape libraries continues to increase, greater requirements are placed on the precision of mechanical positioning in order to successfully read and write data bits. The location of the read/write head in the direction across the tape's width (termed the lateral direction) is actively controlled in order to maintain alignment between the head and data tracks, even in the presence of the tape's lateral vibration. However, during repositioning, vibration is undesirably transmitted from the laterally moving head structure to the axially moving tape because of frictional contact between the two adjacent surfaces. As an analog of that interaction, a model is developed here to describe frictional vibration transmission from a surface having prescribed lateral motion to a tensioned beam that travels and slides over it. For a transport speed that is high when compared to the lateral vibration velocity, Coulomb friction between the surface and the beam can be well-approximated by an equivalent form of viscous damping. The beam is divided into contiguous regions corresponding to free spans and the beam's portion that contacts the surface. A critical engagement length between the beam and the surface exists for which vibration transmission at a particular natural frequency can be substantially reduced, and for a given mode, that length depends weakly on the surface's position along the beam's span. By contouring the surface to have portions of differing radii of curvature, the extent of vibration transmission can be reduced over a broad range of frequency.     

Random vibration of a beam impacting stops

The random vibration of a beam impacting a spring-like stop is discussed. The mean square response and the frequency of impacts are obtained by an equivalent linearization. Reasonable agreement is obtained between these results and the results for an equivalent non-linear single-degree-of-freedom system.     

Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.     

Vibrations of a beam with an absorber consisting of a viscoelastic beam and a spring-viscous damper

The effectiveness of the dynamic vibration absorber which consists of a double-cantilever viscoelastic beam and a spring-viscous damper is studied. The absorber is attached to the centre of the main beam. The ends of the main beam are built in and excited sinusoidally by the base motion. In the numerical example, the displacement transmissibility, i.e., the ratio of the displacement at the centre of the main beam to that of the base, is investigated. Variations of the resonant peaks are shown when the absorber parameters are changed. Values of the optimum tuning design parameters are presented, and it is verified that two of the main beam resonances are optimized simultaneously.     

Measurement of laser light thomson-scattered from a cooling electron beam

First results are presented from an experiment scattering laser light from a relativistic electron beam. The 5 cm diameter continuous electron beam of 28 keV kinetic energy and 2.6 A current presents an electron gas of a density of 8×10⁷ cm^–3, from which 20 ns pulses of laser light (490 nm) were scattered at a repetition rate of 15 Hz and an average power of 20 mJ per pulse. The Doppler-shifted wavelength of photons backscattered under 180° was analysed with a Fabry-Perot interferometer. This technique provides, for the first time, a non-destructive measurement of the velocity distribution in an electron beam radially resolved in space. The results presented here comprise the direct measurement of the absolute electron energy and the degree of space-charge compensation in the electron beam. The determination of an upper bound of 10^–2 for the ratio of longitudinal to transverse electron temperature implies the first direct measurement of a flattened velocity distribution.     

Dynamic characterization of high damping viscoelastic materials from vibration test data

The numerical analysis and design of structural systems involving viscoelastic damping materials require knowledge of material properties and proper mathematical models. A new inverse method for the dynamic characterization of high damping and strong frequency-dependent viscoelastic materials from vibration test data measured by forced vibration tests with resonance is presented. Classical material parameter extraction methods are reviewed; their accuracy for characterizing high damping materials is discussed; and the bases of the new analysis method are detailed. The proposed inverse method minimizes the residue between the experimental and theoretical dynamic response at certain discrete frequencies selected by the user in order to identify the parameters of the material constitutive model. Thus, the material properties are identified in the whole bandwidth under study and not just at resonances. Moreover, the use of control frequencies makes the method insensitive to experimental noise and the efficiency is notably enhanced. Therefore, the number of tests required is drastically reduced and the overall process is carried out faster and more accurately. The effectiveness of the proposed method is demonstrated with the characterization of a CLD (constrained layer damping) cantilever beam. First, the elastic properties of the constraining layers are identified from the dynamic response of a metallic cantilever beam. Then, the viscoelastic properties of the core, represented by a four-parameter fractional derivative model, are identified from the dynamic response of a CLD cantilever beam.     

Analysis of a compliantly suspended acoustic velocity sensor

The dynamics of a compliantly suspended acoustic velocity sensor having a spherical geometry are analyzed using theory and experiment. The analysis starts with a review of the motion associated with an unconstrained solid sphere when subjected to an acoustic plane wave in an unbounded inviscid fluid medium. The theory is then modified to account for the inclusion of an inertial sensor and an external suspension system. Accordingly, the open-circuit receiving response of a geophone-based and accelerometer-based device is derived. Density variations associated with the sphere and the surrounding fluid medium are assessed along with the effects fluid viscosity. Wave effects in the sphere and the suspension system are also analyzed.     

Measurement of the diameter of a laser beam

A new and simple technique for measuring the effective diameter of a laser beam used in material processing is described. The time for the temperature of a spot heated by the laser beam to rise to 90% of equilibrium is compared with that predicted theoretically for a Gaussian TEM₀₀ laser beam. A Gaussian beam diameter equivalent is thus deduced. This calculated diameter is of particular relevance to applications where the laser is used as a heat source.     

Dispersion curves for a viscoelastic Timoshenko beam with fractional derivatives

The Kramers–Kronig dispersion relation, often used as a viscoelastic constitutive law for polymeric materials, is based on purely mathematical properties of linearity, convergence of improper integrals, and causality; thus, it may also be valid as a viscoelastic constitutive law for general structural materials. Accordingly, the motion equation of a Timoshenko beam composed of conventional elastic structural materials is extended to one composed of viscoelastic materials. From the derived governing equation, a dispersive equation is derived for a viscoelastic Timoshenko beam. By plotting phase velocity curves and group velocity curves for a beam of solid circular cross-section composed of a viscoelastic material (polyvinyl chloride foam), the influence of the fractional order of viscoelasticity is examined. As a result, it is found that, in the high frequency range, only the first mode of a Timoshenko beam converged to the propagation velocity of the Rayleigh wave, which takes account of the fractional order of viscoelasticity. In addition, the phase velocity and the group velocity were found to increase as the fractional order approaches 0, and to decrease as the fractional order approaches 1. Furthermore, the rate of velocity change becomes greater as the fractional order approaches 0, and becomes smaller as the fractional order approaches 1.     

Autoparametric vibration absorber effect to reduce the first symmetric mode vibration of a curved beam/panel

This paper presents the implementation of autoparametric phenomena to reduce the symmetrical vibration of a curved beam/panel under external harmonic excitation. The internal energy transfer of a first symmetric mode into first anti-symmetric mode in a curved panel is one example of autoparametric vibration absorber effect. This is similar to the vibration energy transfer from the resonance of a primary structure to the resonance of a secondary spring–mass (tuned mass damper). The nonlinear response of a curved beam is analyzed using an equation with two modes, and a shaker test. The effect of different configurations of the curve beam/panel, including damping ratios and excitation levels, on the energy transfer of the first symmetric mode to the first anti-symmetric mode was studied.The conventional tuned mass damper (TMD) can reduce the resonance response by energy transfer using damping dissipation, whereas an autoparametric vibration absorber (AVA) can reduce the resonance response by energy transfer using parametric interaction. The results indicate that there is a non-absorption region in which vibration is amplified. For the AVA, the non-absorption region can be minimized by tuning the resonance frequency of the first anti-symmetric mode to half of the first symmetric mode resonance frequency using additional mass. No additional damping material is required for achieving sufficient vibration reduction. The AVA can maintain reliable performance in hot and corrosive environments where damping material cannot perform effectively. This paper presents the first successful experimental results of an autoparametric vibration absorption mechanism in a curved beam.     

On the vibration and stability of a free-free beam subjected to end-loads

In this paper the vibration and stability of a free-free beam subjected to direction-controlled axial loads at its ends are investigated. The eigencurves and mode shapes of the beam are presented for various values of the directional control parameter. It is found that the behaviour of the free-free beam subjected to compressive axial loads is unstable for any direction parameter—except for the follower loading case. However, the same beam subjected to tensile loads is stable.