Natural vibration analysis of rail track as a system of elastically coupled beam structures on Winkler foundation

An analytical model for analyzing the vertical free vibration of a rail track is presented. The track structure is represented as a system of elastically coupled beam structures resting on a Winkler foundation. The rail and the tie beams are described by any combination of the two existing beam theories, the Bernoulli-Euler type, and the Timoshenko type, while the rail is assumed to be periodically supported at discrete points on cross-track tie beams. A generalized track element, which consists of a rail span (beam segment), two adjacent ties, and the coupling spring stiffnesses, is established to discretize the track system into identical units. A concept of an equivalent frequency-dependent spring coefficient for the rail support system is introduced to formulate the dynamic stiffness matrix of the track element. Solutions are provided for the natural frequencies of the track and the associated mode shapes of the rail and the ties under transversely (cross-track) symmetric vibration. The free vibration results are used to obtain the dynamic receptance response of a typical field track and to compare them with an existing model and field experimental data.     

Microgimbal torsion beam design using open, thin-walled crosssections

A thin-film micromolding process enabled the construction of microtorsional springs with unique cross-sectional designs by combining high-aspect-ratio beams with horizontal surface features. Cross sections such as T-bars, pi sections, and channels were utilized in creating torsional springs with low torsional stiffnesses and high in- and out-of-plane bending stiffnesses. Experimental modal analysis was used to determine torsional stiffnesses as low as 0.13 μN·m/deg with T-bar springs 45 μm tall, 50 μm wide, and 100 μm long. Springs of the same outer dimensions but with solid rectangular cross sections were calculated to have torsional stiffnesses of at least two orders of magnitude greater. Several microgimbals were constructed using the thin-film micromolding process with various torsional spring designs. Modal analysis was used to experimentally determine pitch and roll natural frequencies. Torsional stiffness models for open, thin-walled sections that included warping effects were developed and used to analytically predict the torsional natural frequencies of tested spring designs to within 20%     

Reality of virtual damage identification based on neural networks and vibration analysis of a damaged bridge under a moving vehicle

Presented here is a reality of virtual damage detection and vibration behaviour study of a discrete beam-like bridge with one or several non-propagating edge cracks subjected to a moving vehicle. In this model, the simply supported beam elements are replaced by a range of rigid bars, which are connected by transverse and rotational springs, while the mass and rotational moment of inertia may be lumped at various points along the beam. The adopted vehicle model here is a four degrees-of-freedom, two axes half-vehicle model with tires flexibility and linear suspensions. Damage can be modelled by altering the spring stiffness equation at the crack position according to predictions, which allows the inclusion of simple or complex damage. To simplify, damage is represented here by an open crack, and stiffness of a given element with damage is calculated by fracture mechanics. Both the discrete element and finite element methods are used to investigate vibration analysis of a discrete beam model subjected to a moving vehicle to confirm model feasibility in vibration analysis under a moving vehicle. Besides, some dynamic response laws are obtained. Considering an irregular road profile, the effects of the moving vehicle velocity, the moving vehicle mass, the crack location and the crack depth on dynamic response of a beam-like bridge are analysed by a numerical example, combining a vehicle–bridge coupled vibration MATLAB program with ANSYS. In addition, the neural network is used to identify the damage of the structure. Numerical results of the numerical model predictions, compared with those obtained from the continuous elements beam, support the accuracy of the discrete elements beam model in both cases of undamaged beam and damaged one. The evidence for condition assessment and damage identification of bridge is obtained from this simulation as obtaining the vibrational characteristics of the damaged beam structure subjected to a moving vehicle. And the inversion results show that the neural network method can identify the injury location and injury size of the structure accurately.

     

直角复合梁压电俘能器多方向能量收集

提出了一种可进行多方向能量收集的直角复合梁压电俘能器。使用有限元分析法得到了直角复合梁压电俘能器的功率频响曲线,并与常规复合梁压电俘能器进行了发电能力比较研究,随后参数化分析了金属主梁长度和附加质量对直角复合梁输出功率的影响关系。结果表明,相比于常规的复合梁,直角复合梁压电俘能器可以进一步减小两个谐振频率之间的距离,并且在两个激励方向（x向和z向）有较高的峰值功率密度。x方向激励时,通过合理调节金属主梁长度和末端质量,直角复合梁压电俘能器可以得到两个更接近的峰值功率密度;z方向激励时,随着金属主梁长度和末端质量的增加,直角复合梁压电俘能器的峰值功率密度增加。     

运动车辆梁模型的横向振动频率及模态

将运动车辆的车身模型化为Eder-Bernoulli梁,车轮模型化为梁两端边界处的弹性不等的弹簧,形成半车模型.通过复模态分析法研究平滑路面上移动车体的横向振动特性,给出车体横向振动的频率方程以及模态的表达式,通过数值方法求解系统固有频率以及模态函数.并通过数值算例研究车辆运行速度、车体刚度、轮胎弹性系数对车体横向振动...     

On the optimal design of the shape of a buckled elastic beam

Analysis of stability, post-buckling bending and vibrations is performed for a beam (a spring element) having an optimal shape. A buckled pin-jointed spring element of a constant thickness and variable width is considered. The optimal shape of this beam is suggested to provide a uniform distribution of maximum bending stresses in its buckled equilibrium configuration for a given value of a supercritical axial force. Sensitivities of a critical force and a buckling mode to variations of the shape of a beam are calculated. A dependence of the static lateral deflection upon an axial force is analysed. Nonlinear equations of large-amplitude oscillations are derived by a use of the Hamilton principle. The natural frequencies of a spring element, compressed by a supercritical force are calculated. Received April 29, 1999     

Natural frequencies of vibration of cantilever sandwich beams

Theoretical and experimental studies were made in obtaining the natural frequencies of cantilever sandwich beams subjected to only gravity forces. The method of minimizing the total energy of the system was used for determining the frequencies. A vibration system made by Unholtz-Dickie was utilized to set the beam in vibration. Resonance occurred when the frequency of the shaker coincided with the natural frequency of the beam. The resonance frequencies were measured by transducers mounted at various locations on the beam. A total of sixteen beams of various lengths, thickness and core density were tested.

It was found that the natural frequency of a cantilever sandwich beam depends largely upon the thickness, length, core density and stiffness of the beam. In addition, the natural frequency has a nonlinear variation with the mode and for any particular mode, the value of the frequency increases as the length of the beam decreases.

Design factors were developed based upon the ratios of the theoretical frequencies of homogeneous beams having the same thicknesses and stiffnesses of that of sandwich beams and of the frequencies experimentally determined for similar sandwich beams.     

VHF free-free beam high-Q micromechanical resonators

Free-free-beam flexural-mode micromechanical resonators utilizing nonintrusive supports to achieve measured Qs as high as 8400 at VHF frequencies from 30 to 90 MHz are demonstrated in a polysilicon surface micromachining technology. The microresonators feature torsional-mode support springs that effectively isolate the resonator beam from its anchors via quarter-wavelength impedance transformations, minimizing anchor dissipation and allowing these resonators to achieve high-Q with high stiffness in the VHF frequency range. The free-free-beam micromechanical resonators of this paper are shown to have an order of magnitude higher Q than clamped-clamped-beam versions with comparable stiffnesses     

Finite element free vibration analysis of damped stiffened panels

Free vibration characleristics of a damped stiffened panel with applied viscoelastic damping on the flanges of the stiffeners are studied using finite element method. The complex nature of the rotational and transverse stiffnesses of the stringers is taken into consideration while deriving the stiffness and mass matrices of the damped stiffener element. The finite element method consists of representing the panel by rectangular plate elements of 12 d.o.f. and the stiffeners by beam elements of 8 d.o.f. which allow for bending, torsional and warping effects. Numerical results showing the effect of the geometric and material properties of the damping layer treatment on the resonant frequencies and loss factors of the composite panel are presented.     

受轴向激励弹性支承梁的稳定性分析

对于广泛存在的弹性支撑梁,首次呈现支承弹簧刚度对轴向激励下梁横向振动稳定性的影响.应用Hamilton原理,建立了两端由线性弹簧支撑的受轴向激励梁的动力学控制方程.通过解析方法计算了受轴向压力梁的固有频率,得到了支撑弹簧刚度与系统固有频率和临界轴力的关系.Galerkin截断后,通过多尺度法和Runge-Kutta法,计算得到了梁参激振动稳态响应的半解析与数值解.讨论了激励幅值、支撑弹簧刚度、平均轴力对系统非线性响应幅值及软硬特性的影响.利用Routh-Hurwitz稳定性判据,求得系统的参激稳定边界,着重讨论了支撑弹簧刚度、阻尼系数的影响.研究发现,边界支撑弹簧的刚度可以显著改变受轴向激励梁的参激稳定边界.因此,研究结果将为广泛存在受到轴向激励结构的设计提供指导.     

Parametric instability of a non-prismatic bar with localized damage subjected to an intermediate periodic axial load

The dynamic instability behaviour of a tapered beam with localized zones of damage and subjected to concentrated periodic loads applied at some intermediate point has been studied by using a finite element analysis. The effects of parameters such as extent and position of damage, position of load, the static and dynamic load factors on the vibration and instability behaviour are discussed. The influence of damage on natural frequency is observed to be most intense at some critical position on the beam. The parametric instability regions are influenced by the combined effects of the position of damage and the position of intermediate load on the beam.     

Hybrid stiff-string–polynomial basis functions for vibration analysis of high speed rotating beams

A new finite element is developed for free vibration analysis of high speed rotating beams using basis functions which use a linear combination of the solution of the governing static differential equation of a stiff-string and a cubic polynomial. These new shape functions depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. The natural frequencies predicted by the proposed element are compared with an element with stiff-string, cubic polynomial and quintic polynomial shape functions. It is found that the new element exhibits superior convergence compared to the other basis functions.     

船舶推进轴系校中对轴系振动影响分析

随着船舶振动噪声要求的提高,现有的静态校中设计方法不再适用,需要考虑轴系校中过程中不对中量对轴系振动的影响.通过对弹性联轴器处三种不对中型式进行受力分析,获得了不对中激励力数学模型,通过台架试验验证了数学模型的准确性.研究表明:轴系不中激励作用下,1倍频和2倍频以及通频振动计算结果与台架试验相对误差小于20%;校中过程中弹性联轴节处不对中量绝对值越大,所产生的激励的幅值越大,造成的振动也越大.     

Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns

An improved numerical method to exactly evaluate 14 × 14 dynamic and static element stiffness matrices is proposed for the spatial free vibration and stability analysis of nonsymmetric thin-walled straight beams subjected to eccentrically axial loads. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a uniform beam element with nonsymmetric thin-walled cross-section. Next a system of linear algebraic equations with nonsymmetric matrices is constructed by introducing 14 displacement parameters and transforming the higher order simultaneous differential equation into the first order simultaneous equation. And then explicit expressions for displacement parameters are exactly evaluated by solving a generalized eigenproblem with complex eigenvalues. Finally exact element stiffness matrices are determined using force-deformation relations. Particularly straightforward application of the present method may not give the exact static stiffness because of existence of multiple zero eigenvalues in case of static buckling problems. Accordingly, a modified numerical method to resolve this difficulty is developed for two cases depending on the initial state of stress resultants. In order to demonstrate the validity and the accuracy of this method, the natural frequencies and buckling loads of nonsymmetric thin-walled beam-columns having bending-torsional deformation modes are evaluated and compared with analytical and F.E. solutions or results analyzed by ABAQUS’s shell element.     

The analysis of nonlinear cable net systems and their supporting structures

The analysis of large cable net systems is feasible only with the aid of high speed digital computers. However, even computerized design of cable nets requires careful formulation and the selection of efficient solution techniques. This paper describes the solution of a general integrated structural system which includes three-dimensional beam members and cable elements that is useful to the design engineer. This solution is used to analyze an example prestressed cable structure, and results are presented to demonstrate the efficiency and accuracy of different solution techniques and to illustrate the effect of variation of important parameters.NET 3 is the general solution program. Input includes: (1) cable sizes, weights, and stress-strain characteristics; (2) initial coordinate points and cable forces due to prestressing; (3) support conditions for the cable net; (4) applied loads; and (5) data for supporting structure. The initial shape and cable forces are computed with the aid of an auxilliary program, SHAPE, given the coordinates of the cable end points, the initial prestressing forces, and an initial interior coordinate. The cable stress-strain characteristics may be linear or nonlinear. The supporting structure may be replaced with equivalent spring stiffnesses or actual three-dimensional elements.The assumptions used in the solution are: (1) cable elements are straight between nodal points and have no flexural stiffnesses; (2) the roofing and decking provide no stiffness; (3) all loads are applied at the nodal points; and (4) all supporting structural elements are elastic. It should be noted that no displacements of the cable nodal points are neglected. Even at boundaries, the elastic stiffness can be considered. Temperature changes can be analyzed to determine their effect upon the structure.Cable net structures usually have many elements and require relatively large amounts of computer time. In addition many loading and geometric conditions must be considered. To be useful as a practical design tool, it was essential that the program be as efficient as possible and require a minimum of storage. The solution utilizes a variation of the Newton-Raphson method. Initially the tangent-modulus stiffness is used. At each iteration the change in the element force is compared with the change in element force for the previous iteration to determine if the convergence rate is within a prescribed value. If it is acceptable, the stiffness for the previous iteration is used; if unacceptable for any element, tangent stiffnesses for all elements are recalculated. This technique attempts to combine the best features of the Newton-Raphson and the modified Newton-Raphson methods by increasing convergence probability and decreasing solution time.To minimize storage requirements, NET 3 has the capability of utilizing tapes in the problem solution. The storage requirements are determined internally before the solution is initiated. If the required storage exceeds that alloted for the system of equations, the solution is effected through the use of tapes.Output for the cables includes nodal point displacements, nodal point forces, and element forces; for beam elements, forces and moments are given. The solution has been verified by comparison with results by other investigators.An elliptical shaped structure, 220 by 240 ft, has been analyzed with the program to investigate the influence of several parameters. The short direction cables were prestressed, and a uniform load of 40 psf was assumed. The variables investigated include: (1) initial prestress force; (2) degree of stiffness of supporting structure; (3) cable sizes; (4) sag-span ratio; and (5) temperature change.     

Longitudinal vibration of carbon nanotubes with elastically restrained ends using doublet mechanics

Free axial vibration analysis of axially restrained carbon nanotubes (CNTs) is studied within the framework of doublet mechanics theory. Fourier sine series are utilized for describing the axial deflection of the carbon nanotube. An eigenvalue approximation is constructed for vibrational modes with the aid of Stokes’ transformation to deformable axial springs. This unclassified approximation bridges the gap between the deformable and rigid boundary conditions. The comparison studies are carried out to verify the efficiency and accuracy of the proposed analytical model by assigning proper values to elastic spring coefficients. The results indicate that the axial springs and small scale parameter of carbon nanotube have considerable effects on the axial vibration behavior of NTs. Similarly, the dependencies of the vibration frequencies on material scale parameter and axial restraints are significant. Similar higher order effects are predicted for other nano or micro structures, all of that confirmed the smaller is stiffer phenomenon.

     

A method for system identification using the optimality criteria optimization approach

An algorithm similar to the optimality criteria approach used in structural optimization is presented for identifying stiffnesses of structural members by using vibration test data. A set of equivalent static inertia forces are obtained from the vibration analysis using d'Alembert's principle and are used to solve the multiple displacement constraint problem. The displacement constraint values are specified based on the measured experimental modal displacement data at critical locations. The algorithm is used to find the changes needed in the stiffnesses of the elements and the distribution of nonstructural mass of the nominal analytical model to correlate the analytical and experimental data. The algorithm alternates between the vibration analysis and static analysis to find the equivalent load vector and modify the stiffnesses. The identified stiffness properties of the structural elements can be used to control and study the dynamic response of the structure.     

Dissipative particle dynamics simulation of shear flow in a microchannel with a deformable membrane

Thin deformable membranes are encountered in a number of microfluidics-based applications. These are often employed for enhancing sorting, mixing, cross-diffusion transport, etc. Microfluidic systems with deformable membranes can be better understood by employing simple models and efficient computational procedures. In this paper, we present a dissipative particle dynamics model to simulate the interaction between a deformable membrane and fluid flow in a two-dimensional microchannel. The membrane is modeled as a bead-spring system with both extensional and torsional springs to simulate extensional stiffness and bending rigidity, respectively. By performing detailed simulations on a membrane pinned at both ends and oriented parallel to the flow, we observe different steady state conformations. These membrane deflections are found to be relatively large for low bending stiffnesses and small for high stiffnesses. The membrane was found to exhibit a simple bowing out mode for high stiffness values and more complex conformations at lower stiffnesses.     

Modelling of double chord rectangular hollow section T-joints by finite element method

A linear elastic T-joint comprised of double chord RHS has been modelled by treating the mated flanges as thin plates supported by coupled linear springs thus simulating the action of the side walls and connecting bottom flanges. A rigid rectangular inclusion is presumed for the branch member. Two loading conditions are analyzed—branch member axial force and branch member bending moment. The finite element formulation that is proposed incorporates rectangular plate and edge boundary spring elements. The model is then used to determine the punching shear and rotational stiffnesses of both double chord T-joints and single chord T-joints, thus demonstrating its versatility. The numerical values obtained are in good agreement with the experimental results available in the literature.