Evaluation of frequency dependent rubber mount stiffness and damping by impact test

A simple experimental method is presented in this paper to evaluate the frequency dependent rubber mount stiffness and damping characteristics by utilizing the measured complex frequency response function from impact test and by least-squares polynomial curve fitting the data obtained from the test. The study shows the transition of the rubber mount stiffness from static to dynamic values. Using the experimentally estimated values of the rubber mount stiffness and damping, the dynamic response of the tested spring-mass system using a rubber mount as the elastic element can be accurately reproduced. In contrast, it is found that the single degree of freedom ideal spring-mass model using constant stiffness and damping values can only predict the response of the system accurately at resonance but not at non-resonance frequencies. The proposed method is validated by comparing its results with those obtained by using mechanical shaker excitations and those of conventional direct stiffness method using blocked transfer frequency response functions.     

复测法测定气轨的粘性阻尼常量

运用解决天平不等臂的复称法的思想，设计了复测法测定气轨的粘性阻尼常量的方法。     

Effect of track stiffness on vibration levels in railway tunnels

The paper discusses the difficulties in comparing and drawing conclusions from vibration measurements made in different railway tunnels. A number of new measurements are presented, which have been made and analyzed on a consistent basis. The track stiffness has been determined for each of the measurements. Graphs are presented which indicate the level of rail and tunnel floor vibration likely to occur for different track stiffnesses. The variance in the measurement results also indicates the potential error in using the data to predict vibration levels in new situations.     

A tuned damping device for reducing noise from railway track

A promising means to reduce the component of railway rolling noise radiated by the track is to increase the damping of the rail. This increases the attenuation with distance of vibrations transmitted along the rail and thereby reduces the noise radiated. To achieve this, a tuned, damped mass-spring absorber system has been designed. To cover a wide range of frequencies, multiple tuning frequencies are used along with a material with a high damping loss factor. Suitable materials have been found from extensive tests on samples and prototypes of the damper have been built and tested, both in the laboratory and in the field. Results are very promising with reductions of the track component of noise of around 6 dB being measured.     

Extreme damping in composite materials with a negative stiffness phase

Composites with negative stiffness inclusions in a viscoelastic matrix are shown to have higher stiffness and mechanical damping tandelta than that of either constituent and exceeding conventional bounds. The causal mechanism is a greater deformation in and near the inclusions than the composite as a whole. Though a block of negative stiffness is unstable, negative stiffness inclusions in a composite can be stabilized by the surrounding matrix. Such inclusions may be made from single domains of ferroelastic material below its phase transition temperature or from prebuckled lumped elements.     

Importance of tread inertia and damping on the tyre/road contact stiffness

Predicting tyre/road interaction processes like roughness excitation, stick-slip, stick-snap, wear and traction requires detailed information about the road surface, the tyre dynamics and the local deformation of the tread at the interface. Aspects of inertia and damping when the tread is locally deformed are often neglected in many existing tyre/road interaction models. The objective of this paper is to study how the dynamic features of the tread affect contact forces and contact stiffness during local deformation. This is done by simulating the detailed contact between an elastic layer and a rough road surface using a previously developed numerical time domain contact model. Road roughness on length scales smaller than the discretisation scale is included by the addition of nonlinear contact springs between each pair of contact elements. The dynamic case, with an elastic layer impulse response extending in time, is compared with the case where the corresponding quasi-static response is used. Results highlight the difficulty of estimating a constant contact stiffness as it increases during the indentation process between the elastic layer and the rough road surface. The stiffness–indentation relation additionally depends on how rapidly the contact develops; a faster process gives a stiffer contact. Material properties like loss factor and density also alter the contact development. This work implies that dynamic properties of the local tread deformation may be of importance when simulating contact details during normal tyre/road interaction conditions. There are however indications that the significant effect of damping could approximately be included as an increased stiffness in a quasi-static tread model.     

Identifying time-dependent damping and stiffness functions by a simple and yet accurate method

The inverse vibration problem is a mathematical process to determine unknown mechanical parameters from measured vibration data. In this study the data of displacement are chosen in order to identify a time-dependent function of damping or stiffness. However, when both functions are to be identified we require both the data of displacement and velocity. This is the first time that a closed-form estimation method for the inverse vibration problems of estimating time-dependent parameters has been constructed. We are able to transform the inverse vibration problem into an identification problem governed by a parabolic-type partial differential equation (PDE). Then, a one-step group-preserving scheme (GPS) for the semi-discretization of PDE is established, which can be used to derive a closed-form solution for estimating parameters. The new Lie-group estimation method has three further advantages: it does not require any prior information on the functional forms of unknown functions; no initial guesses are required; and no iterations are required. Numerical examples were examined to show that the present approach is highly accurate and efficient even for identifying discontinuous and oscillatory parameters. Against the noise is good when only one function is estimated; however, the present approach is slightly weak against the noise when both functions are identified.     

Design of vibration isolators by exploiting the beneficial effects of stiffness and damping nonlinearities

The present study is concerned with the design of a new type of single degree of freedom (sdof) nonlinear vibration isolation system that can deal with harmonic excitations and take advantage of both spring and damping nonlinearities. For typical design requirements expressed in terms of a transmissibility envelope, the proposed design makes use of a recently developed method called the output frequency response function (OFRF) approach, which provides a direct relationship between the system output frequency response and parameters that define the system nonlinearity. Taking all output harmonics into account, a detailed step-by-step procedure is developed to systematically determine the nonlinear parameters from a small set of simulation or experimental data. Simulation studies are conducted to verify the results and demonstrate that the design can effectively achieve all the three requirements for a vibration isolation system of a low resonant peak, low high frequency transmissibility, and a large isolation range.     

Identification of stiffness and damping properties of plates by using the local equation of motion

This paper deals with the identification of stiffness and damping properties of vibrating structures by an inverse method inspired from the Force Analysis Technique (FAT). The proposed approach uses a local equation of motion assumed a priori, which provides a relative straightforward relationship between the displacement field and material properties. The spatial derivatives of the displacement in the equation are calculated using finite differences. As this operation amplifies measurement noise, a regularization step is applied before solving the inverse problem. A procedure is proposed to automatically adjust the level of regularization. The method also allows one to identify local stiffness and damping on a heterogeneous structure. Illustrations for both homogeneous and heterogeneous cases are shown using simulated and measured displacement fields.     

An alternative to modal analysis for material stiffness and damping identification from vibrating plates

This paper presents an alternative to modal analysis to extract stiffness and damping parameters from thin vibrating plates. Full-field slope measurements are performed through a deflectometry technique on a plate vibrating at a given frequency. Images are recorded in phase and at π/2 lag from the excitation. From this information, deflection fields are computed by integration and curvature fields are obtained by differentiation. This information is then input into the principle of virtual work to extract both stiffness and damping parameters. This procedure, known as the Virtual Fields Method, is detailed in the paper and the notion of special optimized virtual fields is extended to the present problem. Validation on simulated data is performed before moving to experimental data. One of the main advantages of this technique is that it is completely insensitive to the damping coming from the boundary conditions. This is illustrated experimentally on two tests where a viscoelastic layer and rubber washers are added in the experimental set up.     

Vibration of a beam on continuous elastic foundation with nonhomogeneous stiffness and damping under a harmonically excited mass

In this paper, a method of analysis of a beam that is continuously supported on a linear nonhomogeneous elastic foundation and subjected to a harmonically excited mass is presented. The solution is obtained by decomposing the nonhomogeneous foundation properties and the beam displacement response into double Fourier summations which are solved in the frequency–wavenumber domain, from which the space–time domain response can be obtained. The method is applied to railway tracks with step variation in foundation properties. The validity of this method is checked, through examples, against existing methods for both homogeneous and nonhomogeneous foundation parameters. The effect of inhomogeneity and the magnitude of the mass are also investigated. It is found that a step variation in foundation properties leads to a reduction in the beam displacement and an increase in the resonance frequency for increasing step change, with the reverse occurring for decreasing step change. Furthermore, a beam on nonhomogeneous foundation may exhibit multiple resonances corresponding to the foundation stiffness of individual sections, as the mass moves through the respective sections along the beam.     

An approach for including the stiffness and damping of elastohydrodynamic point contacts in deep groove ball bearing equilibrium models

This work presents a methodology for including the Elastohydrodynamic (EHD) film effects to a lateral vibration model of a deep groove ball bearing by using a novel approximation for the EHD contacts by a set of equivalent nonlinear spring and viscous damper. The fitting of the equivalent contact model used the results of a transient multi-level finite difference EHD algorithm to adjust the dynamic parameters. The comparison between the approximated model and the finite difference simulated results showed a suitable representation of the stationary and dynamic contact behaviors. The linear damping hypothesis could be shown as a rough representation of the actual hysteretic behavior of the EHD contact. Nevertheless, the overall accuracy of the model was not impaired by the use of such approximation. Further on, the inclusion of the equivalent EHD contact model is equated for both the restoring and the dissipative components of the bearing?s lateral dynamics. The derived model was used to investigate the effects of the rolling element bearing lubrication on the vibration response of a rotor?s lumped parameter model. The fluid film stiffening effect, previously only observable by experimentation, could be quantified using the proposed model, as well as the portion of the bearing damping provided by the EHD fluid film. Results from a laboratory rotor–bearing test rig were used to indirectly validate the proposed contact approximation. A finite element model of the rotor accounting for the lubricated bearing formulation adequately portrayed the frequency content of the bearing orbits observed on the test rig.     

Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method

In this paper, an explicit formulation of the incremental harmonic balance (IHB) scheme for computation of periodic solutions of a harmonically excited oscillator which is asymmetric with both stiffness and viscous damping piecewise linearities is derived. Analysis of dynamical behavior as bifurcation and chaos of the non-linear vibration system considered is effectively carried out by the IHB procedure, showing that the system exhibits chaos via the route of period-doubling bifurcation, with coexistence of multiple periodic attractors observed and analyzed by the interpolated cell mapping method. In addition, numerical simulation by the IHB method is compared with that by the fourth order Runge-Kutta numerical integration routine, which shows that this method is in many respects distinctively advantageous over classical approaches, and especially excels in performing parametric studies.     

Simulation of the track growth and determining the track parameters

We have found the equation of the etch pit wall in solid state nuclear track detectors, as follows:

where: x is the distance along a track from the point where the particle entered the detector; V(x) is the ratio of the track etch rate to the bulk etch rate; C is the integration constant that can be determined from particle penetration depth, and y is the normal distance from the particle trajectory to the etch pit wall. The equation is derived assuming the increasing track etch rate V_t along the particle trajectory.

The above equation can be used for the simulation of the track growth and calculating the major and the minor axis of the etch pit opening. The corresponding computer program was set up. The input parameters of this program are: alpha particle energy, incidence angle and removed layer: the output are track parameters. The results obtained by this method are compared with another approach given by Somogyi and Szalay (1973) and reasonably good agreement is found.     

Vibration and damping analysis of annular plates with constrained damping layer treatments

The natural frequencies and modal loss factors of annular plates with fully and partially constrained damping treatments are considered. The equations of free vibration of the plate including the transverse shear effects are derived by a discrete layer annular finite element method. The extensional and shear moduli of the viscoelastic material layer are described by the complex quantities. Complex eigenvalues are then found numerically, and from these, both frequencies and loss factors are extracted. The effects of viscoelastic layer stiffness and thickness, constraining layer stiffness and thickness, and treatment size on natural frequencies and modal loss factors are presented. Numerical results also show that the longer constrained damping treatment in radial length does not always provide better damping than the shorter ones.     

利用阻尼振动测量气垫导轨阻尼系数的新思路

利用气垫导轨上两根弹簧连接滑块做阻尼振动的模型,通过求解阻尼振动方程推算得到了滑块运动的最大速度衰减式,从而求得气垫导轨的阻尼系数.此外,从品质因数的角度推算阻尼系数的计算式,在阻尼很小时,该方法可视为最大速度衰减法的近似.通过气垫导轨阻尼振动实验,测量振幅、周期和光电门遮光时间等物理量,利用最大速度衰减和品质因数法得出阻尼系数的值非常接近.进一步对做阻尼振动的滑块拍摄视频,通过慢放视频得到其振动过程中振幅的衰减值,利用最大振幅衰减法计算出阻尼系数,所得结果与上述两种方法的数值符合较好,从另一个角度证明了本文提出的最大速度衰减法和品质因数法测阻尼系数的可靠性,该类方法在实验中易于操作、实践性强、所得结论可靠,为气垫导轨阻尼系数的测量提供了一种新的思路.     

Hysteretic damping and causality

It is proven that linear oscillatory systems with hysteretic damping in the form of complex stiffness and/or complex elastic moduli satisfy the causality principle: the response of such a system to an arbitrary external force cannot appear earlier than the onset of the force. The proof, based on a rigorous solution to the problem of forced oscillations, is presented in detail for an oscillator with a complex stiffness, as well as in a brief explanation for a system with N mass. It is also shown that these systems are Lyapunov-unstable. A comparison is made to other linear hysteretic damping models.     

Calibration of PM-355 nuclear track detectors; comparison of track diameter diagrams with track depth characteristics

The calibration diagrams, i.e. track diameters and track depths versus ion energy and etching time, as obtained for PM-355 track detector irradiated with He-ions are presented. The both detector characteristics are compared. The track etch rate is determined by two methods, as a function of the etch pit depth and the ion energy loss.     

Improvement of damping characteristics of structural members with high damping elastic inserts

The major objective of this work is to investigate theoretically the possibility of improving the damping capacities of structural members with the help of inserts of special high damping materials. The inserts have been considered to be both welded and press-fit to the members. It has been observed that the effectiveness of the press-fit inserts is much more than that of the welded inserts. In case of the welded inserts there exists an optimum size whereas for shrink-fit inserts best results are expected when the inserts are solid. It has been found that with such a method the damping capacity of a member can be considerably increased with no significant loss in static rigidity.