Time- and frequency-domain analysis using lumped mass global collocation

Quite recently, a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures was proposed (Archive of Applied Mechanics: DOI: ). This paper extends the latter methodology on several levels, in both the time and frequency domain. Firstly the formulation is updated so that it can also deal with rods of variable cross section. Then, the fully populated mass matrices of the previous formulation are properly replaced by lumped masses, thus saving still more computer effort. Subsequently, a new general formulation for the transient response analysis is proposed. Finally, a novel procedure for the coupling of two neighboring collinear rods is presented. The theory is supported by six test cases concerning elastic rods of constant and variable cross sections. Among these, transient analysis refers to the response of a single rod due to a Heaviside-type loading as well as to the impact between two collinear rods of different cross sections.     

Static analysis of two-dimensional elastic structures using global collocation

Based on previous findings concerning the numerical solution of one-dimensional elastodynamical problems [Provatidis in Arch Appl Mech 78(4):241–250, 2008] this paper extends the methodology to the static analysis of two-dimensional problems in quadrilateral domains. This target is achieved by replacing the Galerkin/Ritz procedure involved in Lagrangian (or Gordon–Coons) type finite elements by a global collocation scheme. In brief, the boundary conditions are fulfilled at all boundary nodes, while the governing equation is fulfilled at internal points. The theory is supported by four test cases concerning rectangular and curvilinear structures under plane-stress or plane-strain conditions, where the convergence rate is successfully compared with that of conventional bilinear finite elements with the same mesh density.     

Free vibration analysis of metal foam core sandwich beams on elastic foundation using Chebyshev collocation method

Archive of Applied Mechanics - In this paper, free vibration of a metal foam core sandwich (MFCS) beam embedded in Winkler–Pasternak elastic foundation is studied using the Chebyshev...     

A meshless collocation method for the plane problems of functionally graded material beams and plates using the DRK interpolation

A meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the differential reproducing kernel (DRK) interpolation, in which the DRK interpolant is constructed by the randomly distributed nodes. A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach with excellent accuracy and has a fast convergence rate.     

Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method

Free vibration analysis of moderately thick rectangular FG plates on elastic foundation with various combinations of simply supported and clamped boundary conditions are studied. Winkler model is considered to describe the reaction of elastic foundation on the plate. Governing equations of motion are obtained based on the Mindlin plate theory. A semi-analytical solution is presented for the governing equations using the extended Kantorovich method together with infinite power series solution. Results are compared and validated with available results in the literature. Effects of elastic foundation, boundary conditions, material, and geometrical parameters on natural frequencies of the FG plates are investigated.     

Free vibration analyses by boundary element method using component mode synthesis

The boundary element method for dynamic analysis of two dimensional structures was developed by Nardini and Brebbia in 1983. Free vibration analysis of structures by BEM results in a generalized eigenproblem with coefficient matrices nonsymmetric and full. So, it costs a lot of CPU time to solve the eigenproblem. In order to reduce the cost the improved free interface coupling method has been adopted. By aid of lower eigenvectors and residual modes the degrees of freedom of substructure equaion of dynamic equilibrium can be reduced. By assembling the reduced substructure equations, the system of equations for the whole structure will be obtained. By using the traction and displacement compatibility conditions on the interface, the second transformation matrix has been developed for further reduction of system equations. The technique has been applied to some specific examples which give the evidence that the method is satisfactory and efficient.     

Comparison of methods for vibration analysis of electrostatic precipitators

The paper presents two methods for the formulation of free vibration analysis of collecting electrodes of precipitators.The first,called the hybrid finite element method, combines the finit element method used for calculations of spring deformations with the rigid finite element method used to reflect mass and geometrical features,which is called the hybrid finite element method.As a result,a model with a diagonal mass matrix is obtained.Due to a specific geometry of the electrodes,which are long plates of complicated shapes,the second method proposed is the strip method which is a semi-analytical method.The strip method allows us to formulate the equations of motion with a considerably smaller number of generalized coordinates.Results of numerical calculations obtained by both methods are compared with those obtained using commercial software like ANSYS and ABAQUS.Good compatibility of results is achieved.     

Large amplitude free flexural vibration of rings using finite element approach

Here, the large amplitude free flexural vibrations of isotropic/laminated orthotropic rings are investigated, using a shear flexible curved beam element based on field consistency principle. A laminated refined beam theory is introduced for developing the element, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the inner and outer surfaces of the beam. The formulation includes in-plane and rotary inertia effects, and the non-linearity due to the finite deformation of the ring. The governing equations obtained using Lagrange's equations of motion are solved through the direct integration technique. Amplitude-frequency relationships evaluated from the dynamic response history are examined. Detailed numerical results are presented considering various parameters such as radius-to-thickness ratio, circumferential wave number and ovality for isotropic and laminated orthotropic rings. The nature and degree of the participation of various modes in non-linear asymmetric vibration of oval ring brought out through the present study are useful for accurate modelling of the closed non-circular structures.     

Static and free vibration analysis of laminated glass beam on viscoelastic supports

Static behavior and free vibration analysis of laminated glass beam on viscoelastic supports are performed. For the static case, an analytical way is developed for analyzing and optimization of laminated glass beam with general restraints at the boundaries. In the case of free linear vibrations, the modal properties of the glass are determined using a finite element method which is a powerful tool in the design of support damping treatment of a sandwich glass for passive vibration control.     

复合材料结构的内部分层及其扩展

本文用接触观点分析了复合材料壳体结构的内部分层问题，建立了求解摩擦接触问题的有限元迭代格式，并采用释放率判据分析分层裂纹的扩展。分层裂纹前缘采用轴对称奇异单元，并了结构的大变形特性。计算表明，层间裂纹呈Ⅱ型扩展趋势。本文同时还分析了层间摩擦系数的影响，并指出分层将导致壳体结构承载能力显著下降。     

A modeling and vibration analysis of a piezoelectric micro-pump diaphragm

The vibration analysis of a micro-pump diaphragm is presented. A piezoelectric micro-pump is studied. For this purpose, a dynamic model of the micro-pump is derived. The micro-pump diaphragm is modeled as circular double membranes, a piezoelectric one as actuator and a silicon one for representing the membrane for pumping action. The damping effect of the fluid is introduced into the equations. Vibration analysis is established by explicitly solving the dynamic model. The natural frequencies and mode shapes are calculated. The orthogonality conditions of the system are discussed. To verify the results, the finite-element micro-pump model is developed in ANSYS software package. The results show that the two methods are well comparable.     

Free vibration of non-uniform column using DQM

Most of engineering problems are governed by a set of partial differential equations with proper boundary conditions. The present work is concerned with free vibration analysis of non-uniform column resting on elastic foundation and subjected to follower force. The used method of solution is the differential quadrature method (DQM). Formulation of the problem is introduced. The results obtained and compared with the exact solution and traditional numerical techniques such as finite element method. The parametric study is used to investigate the effect of column geometry on the natural frequencies, the mode shapes and the critical load.     

Free vibration analysis of thin rotating cylindrical shells using wave propagation approach

The wave propagation approach is extended to study the frequency characteristics of thin rotating cylindrical shells. Based on Sanders’ shell theory, the governing equations of motion, which take into account the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotation, are derived. And, the displacement field is expressed in the form of wave propagation associated with an axial wavenumber k _m and circumferential wavenumber n. Using the wavenumber of an equivalent beam with similar boundary conditions as the cylindrical shell, the axial wavenumber k _m is determined approximately. Then, the relation between the natural frequency with the axial wavenumber and circumferential wavenumber is established, and the traveling wave frequencies corresponding to a certain rotating speed are calculated numerically. To validate the results, comparisons are carried out with some available results of previous studies, and good agreements are observed. Finally, the relative errors induced by the approximation using the axial wavenumber of an equivalent beam are evaluated with respect to different circumferential wavenumbers, length-to-radius ratios as well as thickness-to-radius ratios, and the conditions under which the analysis presented in this paper will be accurate are discussed.     

Post-buckling analysis of slender elastic rods subjected to terminal forces

This paper presents formulation and solutions for the elastica of slender rods subjected to axial terminal forces and boundary conditions assumed hinged and elastically restrained with a rotational spring. The set of five first-order non-linear ordinary differential equations with boundary conditions specified at both ends constitutes a complex two-point boundary value problem. Solutions for buckling, initial post-buckling (perturbation), large loads (asymptotic) and numerical integration are developed. Results are presented in non-dimensional graphs for a range of rotational spring stiffness, tuning the analysis from double-hinged to hinged-built-in rods.     

Free vibration analysis of third-order shear deformable composite beams using dynamic stiffness method

The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.     

Numerical vibration analysis on defect detection in revolving machines using two bearing models

Summary Monitoring by vibration measurement and analysis is largely used in the industry for detection of defects in revolving parts of machines. The determination of good sensor positions is one of the main research goals in the field of predictive maintenance. This paper proposes a numerical methodology based on the FEM and spectral analysis in order to find the optimum sensor positions. Bearings are key components in the vibration propagation from moving parts to immobile ones. Two existing nonlinear models of bearings are recalled and implemented in a FE code. The obtained tangent stiffness matrices of bearings are then put in the global system to-study the dynamic behaviour. The dynamic response of the whole system under defect excitations is used to determine the optimum sensor placements for the defect detection in the predictive maintenance.     

On the constraint and scaling methods to derive the equations of linearly elastic rods

The theory of linearly elastic rods may be obtained from three-dimensional elasticity either by the method of internal constraints or by the scaling method. Both methods have been applied to obtain linear plate and shell equations ([1], [2]–[5]); the relationships between the two methods are discussed in [6]. For rods, a version of the constraint method has been developed in [7], whereas a scaling method has been presented in [12]. In this paper a direct comparison is made between the mechanical basis and analytical results of the constraint and the scaling methods, and it is shown how the scaling method yields the same Kirchhoff hypothesis that forms the starting point of the constraint method.

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Sommario La teoria delle travi elastiche lineari puó essere ottenuta a partire dalla teoria tridimensionale dell'elasticitá tanto con il metodo dei vincoli interni che con il metodo di riscalamento. Entrambi i metodi sono giá stati utilizzati per ottenere le equazioni delle piastre e dei gusci lineari ([1], [2]–[5]); in quel contesto, le relazioni tra i due metodi sono state discusse in [6]. Una versione del metodo dei vincoli appropriata al caso delle travi é stata sviluppata in [7], mentre i metodi di riscalamento per le travi si trovano esposti in [12]. Scopo di questo lavoro é compiere un paragone diretto tra i fondamenti meccanici e le risultanze analitiche, rispettivamente, del metodo dei vincoli e di quello di riscalamento, mostrando come il metodo di riscalamento imponga di accogliere proprio quelle ipotesi all aKirchhoff sulle quali si basa il metodo dei vincoli interni.