Free vibration analysis of sandwich beam carrying sprung masses

In this paper, free vibration of three-layered symmetric sandwich beam carrying sprung masses is investigated using the dynamic stiffness method and the finite element formulation. First the governing partial differential equations of motion for one element are derived using Hamilton’s principle. Closed form analytical solution of these equations is determined. Applying the effect of sprung masses by replacing each sprung mass with an effective spring on the boundary condition of the element, the element dynamic stiffness matrix is developed. These matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the well known Wittrick–Williams algorithm. Free vibration analysis using the finite element method is carried out by increasing one degree of freedom for each sprung mass. Finally, some numerical examples are discussed using the dynamic stiffness method and the finite element formulation. After verification of the present model, the effect of various parameters such as mass and stiffness of the sprung mass is studied on the natural frequencies.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

Derivation of stiffness matrices for modeling a human cervical spine

Based on the published experimental data under static loading condition, stiffness matrices of the cervical spine were developed. These matrices simulate well the static response of the cervical spine as shown by the authors in an earlier work, while for a dynamic simulation these matrices need some modifications in order to achieve more accurate results. We present a method, in the form of a constrained optimization problem, that facilitates the evaluation of the stiffness matrices to be used in a dynamic simulation. The intervertebral disk and the ligaments are represented by an equivalent stiffness matrix whose elements are assumed to be constant over the entire range of mobility. With a set of new stiffness matrices derived by the proposed method, the dynamic simulation showed a good agreement with the observed motion data.     

A study on vibration of Stewart platform-based machine tool table

In this paper, an analytical study on the vibrations of a parallel manipulator is addressed. In the vibration equation of the moving platform, the damping and stiffness of the pods are taken into account. The eigenvalue problem of the moving platform is solved to obtain the natural frequencies. Considering the role of different factors effective on the mass and stiffness matrices of the platform, natural frequencies for different configurations are investigated. The results obtained by analytical approach are further verified through FEM simulation. The effect of variation in position and orientation of the moving platform on the change in stiffness of its supporting chain, inertia tensor and natural frequencies and mode shapes of the platform as well as the effects of different payloads are studied. The vibration of the platform in different configurations is studied in different cutting conditions. The ranges of resonance frequencies and vibration amplitudes are then investigated. Finally, proper configurations of the moving platform are determined to avoid dynamic instability in different machining conditions. It also will be illustrated in this paper that some specific features embodied in the mechanism are appropriate for high-speed milling.     

Exact dynamic stiffness matrix of a Timoshenko three-beam system

An exact dynamic stiffness matrix is established for an elastically connected three-beam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them. The formulation includes the effects of shear deformation and rotary inertia of the beams. The dynamic stiffness matrix is derived by rigorous use of the analytical solutions of the governing differential equations of motion of the three-beam system in free vibration. The use of the dynamic stiffness matrix to study the free vibration characteristics of the three-beam system is demonstrated by applying the Muller root search algorithm. Numerical results for the natural frequencies and mode shapes of the illustrative examples are discussed for 10 interesting boundary conditions and three different stiffness constants of springs.     

Effect of transmission error on the dynamic behaviour of gearbox housing

The dynamic response of gearbox remains a paramount concern because of noise generation. This work is concerned with numerical simulation of the overall dynamic behaviour of a parallel helical gear transmission. A dynamic sub-structuring method using different types of substructure (carrying and slave) is made to determine the natural frequencies and the corresponding mode shapes. The structure to be investigated is subdivided into components or sub-structures, which are then analyzed independently for natural frequencies and mode shapes. A numerical model taking into account the elastic coupling between the various components of a gearbox was developed. It allows studying and analyzing the dynamic behaviour of elastic housing in the presence of gear process. The static transmission error is introduced as a vibratory excitation source and it is represented by time-varying mesh stiffness. The discretization of the housing deformation energy and the kinetic energy expressions using plate finite elements leads to constructions of the stiffness and the mass matrixes. In dynamic analyses, time-discretization based on the Newmark method is used. The different equations governing movement of gearbox are established in a truncated modal base deduced from the average characteristics of the structure. A gearbox example is presented, and analyzed. A presentation and discussion of the numerical results was emphasized. The numerical results allow us to conclude on the dominant phenomena of the overall dynamic behaviour of the gear transmission.     

Roles of domain decomposition method in plate vibrations: Treatment of mixed discontinuous periphery boundaries

The detailed development of a domain decomposition method (DDM) for the vibrational modelling of rectangular plates with mixed-edge boundary conditions is presented. In the DDM, the complex plate domain is decomposed into small simple subdomains and the appropriate shape function of each subdomain is represented by sets of admissible orthogonal polynomials generated using the Gram-Schmidt recurrence process. The continuity matrices that couple the eigenvectors of adjacent subdomains are derived based on the satisfaction of continuity conditions along the interconnecting boundaries. The stiffness and mass matrices of each subdomain after pre- and post-multiplication by the respective continuity matrices are assembled to form the global stiffness and mass matrices. To demonstrate the effectiveness and accuracy of the DDM, a vibration study of several partially mixed edge plates has been carried out. Convergence tests for example problems are presented in which the accuracy of the results is established. The frequency parameters and mode shapes obtained, where possible, are verified by comparison with data published in the open literature.     

The performance characteristics of a micro-machined Coriolis flow meter: An evaluation by simulation

Micro-machined Coriolis meters will enable measurement of very low flow rates (0.1–500 g/h) and, potentially, ultra-low flows (0.1–100 mg/h). Application areas include the delivery of medical drug infusions to patients, and a wide variety of micro-fluidic devices. An evaluation of the performance of two prototype micro-machined flow-tubes of differing shapes is reported, based upon results obtained from a virtual Coriolis meter. This tool comprises a finite element modelling capability which simulates the meter flow-tube in motion, with the flow represented simply as a continuous string, i.e. 1-dimensional and frictionless, and the model allows the generation of pseudo-data at points on the tube corresponding to sensor locations. Application of signal processing algorithms then enables the representation of an indicated flow time history output by a Coriolis meter in response to a prescribed input flow. Results indicate that the devices investigated were all highly linear and that meter sensitivity is independent of fluid density. One flow-tube shape confers higher stiffness than the other and, for both tube shapes, increasing wall thickness increases tube stiffness at a greater rate than the tube mass. Higher stiffness results in reduced meter sensitivity, but increased drive frequency (hence, faster dynamic response). The spatial averaging resulting from the use of ‘distributed’ internal sensors inevitably yields meter sensitivity values that are lower than the potential maximum value that might be achieved by use of ‘point’ sensors; however there are practical reasons why this latter approach would not work. The dynamic response to a flow step is essentially the same as found for macro-Coriolis meters.     

General sinusoidal stiffness matrices for buckling and vibration analyses of thin flat-walled structures

This paper is concerned with the derivation of stiffness matrices for the buckling or vibration analysis of any structure consisting of a series of long, thin, flat plates rigidly connected together at their longitudinal edges. Each plate is assumed to be subjected to a basic state of plane stress which is longitudinally invariant, and it is further assumed that the mode of buckling or vibration varies sinusoidally in the longitudinal direction. During buckling or vibration, the edges of any individual plate are subjected to additional systems of forces and moments which are sinusoidally distributed along the edges, and these give rise to sinusoidally varying edge displacements and rotations. Spatial phase differences between the forces and displacements are accounted for by defining them in terms of complex quantities. The sinusoidal edge forces and displacements are split into two uncoupled systems, corresponding to out-of-plane and in-plane displacements, and two stiffness matrices are defined. The out-of-plane stiffness matrix is shown to be in general complex, and Hermitian in form, but the inplane stiffness matrix is real and symmetrical. Explicit expressions are derived for the elements of the matrices, in which all the essential destabilizing effects of the basic stresses, as well as dynamic effects, are included. Finally, it is shown that buckling and vibration phenomena for any structure of this type are closely interrelated.     

考虑剪切变形的复杂刚架结构动力学特性分析

采用谱元法计算分析了复杂刚架结构的动力学响应特性,建立了直杆和Timoshenko梁的局部动力刚度阵,并组装得到了整体刚架结构的动力刚度阵。计算了刚架结构的频响曲线,并与有限元法得到的结果进行了比较,获得了刚架的固有频率和冲击载荷作用下的频域曲线。研究结果表明,谱元法可有效且准确地分析刚架结构的动力学响应特性,尤其适合中高频动态响应特性分析,与有限元方法相比,其精度高且用时短。       

Use of a generalized beam/spring element to analyze natural vibration of rail track and its application

This paper describes the formulation of a generalized beam/spring track element to obtain the natural vibration characteristics of a railway track modeled as a periodic elastically coupled beam system on a Winkler foundation. The rail/tie beams are described by either the Timoshenko beam theory or the Bernoulli-Euler beam theory. The rail beam is assumed to be discretely coupled to the cross-track ties through the coupling spring elements at the periodic rail/tie intersections. The generalized beam/spring element consists of a rail span beam segment, two adjacent tie beams, the coupling spring elements and the ultimate foundation stiffness. The entire track/beam system is then discretized into an assembly of periodic structural units. An equivalent frequency-dependent spring coefficient representing the resilient, flexural and inertial characteristics of the track substructure unit is formulated to establish the dynamic stiffness matrix of the generalized element. The eigenvalue problem of the track/beam system is solved by employing a comprehensive and efficient numerical routine. Solutions are provided for the natural frequencies of the track and the mode shapes of the rail/tie beams under transversely (cross-track) symmetric vibration. The natural 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.     

Natural frequency and mode shape analysis of structures with uncertainty

Two methods called random factor method (RFM) and interval factor method (IFM) for the natural frequency and mode shape analysis of truss structures with uncertain parameters are presented in this paper. Using the RFM, the structural physical parameters and geometry can be considered as random variables. The structural stiffness and mass matrices can then, respectively, be described by the product of two parts corresponding to the random factors and the deterministic matrix. The structural natural frequencies, mode shapes and random response can be expressed as the function of the random factors. By means of the random variable's algebra synthesis method, the computational expressions for the mean value and standard deviation of natural frequencies and mode shapes are derived from the Rayleigh quotient. Using the IFM, the structural parameters can be considered as interval variables and the computational expressions for the lower and upper bounds of the natural frequency and mode shape are derived by means of the interval operations. The effect of uncertainty of individual structural parameters on structural dynamic characteristics, and the comparison of structural natural frequency and mode shape using the RFM and IFM are demonstrated by truss structures.     

Substructuring and model reduction of pipe components interacting with acoustic fluids

This paper presents a model reduction and substructure technique for reduced dynamical models of fluid-filled pipe components. Both linear acoustical domain and structural domain are modelled by finite elements (FE), and they are fully coupled by a fluid–structure interface. The discretised dynamic FE-equations, which use the acoustic pressure as field variable in the fluid, render both non-symmetric mass and stiffness matrices due to the FSI-coupling. Since the partial solutions to the eigenproblem of the coupled system are of special interest, either numerical preconditioning or non-dimensionalisation of the physical quantities is performed to improve the condition and to accelerate the numerical computation. An iterative subspace solver is adopted to generate a sufficient approximate of the low-frequency eigenspace of the constrained problem. Model reduction for component mode synthesis uses constraint modes together with the computed eigenspace. Single-point constraints for the nodal degrees of freedom hold at the interface between substructures. The null space resulting from a QR-decomposition of the single-point constraints at the interface is used as explicit coupling matrix to prevent the deterioration of the conditioning. Partitioning of the reduction space and coupling matrices leads to a structure of the coupled global system matrices, which is similar to the original system structure in physical quantities. Therefore, the iterative subspace eigensolver is used again for numerical modal analysis. Modal analysis is performed for a pipe segment assembled by fully coupled two-field substructures. The results are compared to the results obtained from the full model and to experimentally determined mode shapes.     

Comparative study of damped FE model updating methods

Most of finite element (FE) model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.     

不确定声场分析的区间矩阵分解摄动有限元法

针对修正一阶区间摄动有限元法存在的一阶Taylor展开误差较大和求解摄动逆矩阵时计算效率不高的缺陷,提出区间矩阵分解摄动有限元法(Decomposed interval matrix perturbation finite element method, DIMPFEM)。该方法将系统动态刚度矩阵分解为若干系统子矩阵之和,每个系统子矩阵的摄动矩阵用摄动因子和常量矩阵的乘积表示,避免了摄动矩阵的Taylor展开误差;采用Epsilon算法求解摄动逆矩阵的修正Neumann级数,有效提高了计算效率。将DIMPFEM应用于具有区间参数的二维管道和二维商务车声腔模型的声压响应分析,分析结果表明,与修正一阶区间摄动有限元法比较,DIMPFEM获得了更高的计算精度和计算效率。     

Analysis of dynamic characteristics of structural joints using stiffness influence coefficients

This paper proposes a new modeling method for joints in mechanical structures in order to reduce the errors in eigenvalue analysis due to joint modeling. The new modeling method uses both a stiffness influence method and a condensation method to obtain the dynamic characteristic matrix of the joint region. It also employs the displacement and reaction of finely modeled finite element analysis in the calculation of stiffness influence coefficients. In order to check the validity of the proposed method, natural frequencies and mode shapes of a simple structure with a bolted joint are investigated by the proposed method and by experiments. The eigenvalue analysis using the proposed method shows more accurate results than that using rigid joints modeling, when the natural frequencies are compared with the experimental results. In addition, the differences between the natural frequencies obtained by the proposed method and those by the rigid joints modeling are notable in the modes where the joint has elastic deformation.     

考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。     

A finite element beam analysis by interface matching

An alternative way of finite element beam analysis is presented. The beam deflection in an element is represented by the sum of general solution and particular solution. The general solution is approximated by using Hermite polynomials and the particular solution is obtained by applying zero boundary conditions at element boundaries. The inter-element stiffness matrices are obtained by requiring the continuity of moment and shear force across element boundaries. The inter-element stiffness matrices do not overlap each other to form the global stiffness matrix. The boundary conditions are explicitly specified. Numerical examples are provided for various boundary conditions and load conditions.

     

Non-linear free vibration analysis of laminated orthotropic cylindrical shells

A method to predict the influence of geometric non-linearities on the natural frequencies of an empty laminated orthotropic cylindrical shell is presented in this paper. It is a hybrid of finite element and classical thin shell theories. Sanders—Koiter non-linear and strain-displacement relations are used. Displacement functions are evaluated using linearized equations of motion. Modal coefficients are then obtained for these displacement functions. Expressions for the mass, linear and non-linear stiffness matrices are derived through the finite element method (in terms of the elements of the elasticity matrix). The uncoupled equations are solved with the help of elliptic functions. The frequency variations are first determined as a function of shell amplitudes and then compared with the results in the literature.     

Flexural-torsional coupled vibration of slewing beams using various types of orthogonal polynomials

Dynamic behavior of flexural-torsional coupled vibration of rotating beams using the Rayleigh-Ritz method with orthogonal polynomials as basis functions is studied. Performance of various orthogonal polynomials is compared to each other in terms of their efficiency and accuracy in determining the required natural frequencies. Orthogonal polynomials and functions studied in the present work are : Legendre, Chebyshev, integrated Legendre, modified Duncan polynomials, the special trigonometric functions used in conjunction with Hermite cubics, and beam characteristic orthogonal polynomials. A total of 5 cases of beam boundary conditions and rotation are studied for their natural frequencies. The obtained natural frequencies and mode shapes are compared to those available in various references and the results for coupled flexural-torsional vibrations are especially compared to both previously available references and with those obtained using NASTRAN finite element package. Among all the examined orthogonal functions, Legendre orthogonal polynomials are the most efficient in overall CPU time, mainly because of ease in performing the integration required for determining the stiffness and mass matrices.