Vibration analysis of plane frames by customized stiffness and diagonal mass matrices

2011 ◽  
Vol 225 (12) ◽  
pp. 2848-2863 ◽  
Author(s):  
M Rezaiee-Pajand ◽  
R Khajavi
Keyword(s):  
Vibration Analysis ◽  
Simple Structure ◽  
Frame Structures ◽  
Numerical Examples ◽  
Beam Elements ◽  
Stiffness Matrices ◽  
Mass Matrices ◽  
Plane Frames ◽  
Accurate Performance ◽  
Vibration Problems

This article presents a formulation for the vibration analysis of plane frames. The strain gradient notation is utilized to determine the mass and stiffness matrices. The obtained matrices can easily be parameterized due to their simple structure. Both Euler-Bernoulli- and Timoshenko-beam elements are investigated in this study. The parameterized stiffness and mass matrices are optimized for accurate performance in the vibration analysis of frame structures. Some numerical examples are solved to show the advantages of the presented scheme. Results of these sample vibration problems indicate that the proposed technique increases the accuracy of analysis, when these new stiffness and diagonal mass matrices are used.

Vibration Analysis of Beams with Arbitrary Elastic Boundary Conditions

2011 ◽  
Vol 66-68 ◽  
pp. 1325-1329
Author(s):  
Bing Lin Lv ◽  
Wan You Li ◽  
Jun Dai ◽  
Hai Jun Zhou ◽  
Fei Xiang Guo ◽  
...  
Keyword(s):  
Fourier Series ◽  
Vibration Analysis ◽  
Elastic Boundary ◽  
Stiffness Matrices ◽  
Fourier Series Method ◽  
Mass Matrices ◽  
Ritz Procedure ◽  
Elastically Supported ◽  
Elastic Boundary Conditions ◽  
Improved Fourier Series Method

In this paper, one newly developed method named the Improved Fourier Series method is applied to the vibration analysis of a beam elastically supported at the both end. The flexural displacement of the beam is supposed to be one set of Fourier Series coupled with four appended terms. Based on the Rayleigh-Ritz procedure and and the vibration characteristics of the beam are also acquired by solving these two matrices. In the end, the frequencies calculated are also compared with those from references and Results ar the Hamilton’s equation, the mass matrices and the stiffness matrices of the beam are obtained e all proved excellent.

Differential Quadrature Element Method for the Analysis of Vibration of Frame Structures Having Nonprismatic Members Considering Warping Torsion

2000 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Vibration Analysis ◽  
Differential Quadrature ◽  
Frame Structures ◽  
Rigorous Solution ◽  
Discrete Eigenvalue ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Domain Boundaries ◽  
Eigenvalue System ◽  
Element Method

Abstract The differential quadrature element method (DQEM) and the extended differential quadrature (EDQ) have been proposed by the author. The EDQ is used to the DQEM vibration analysis frame structures. The element can be a nonprismatic beam considering the warping due to torsion. The EDQ technique is used to discretize the element-based differential eigenvalue equations, the transition conditions at joints and the boundary conditions on domain boundaries. An overall discrete eigenvalue system can be obtained by assembling all of the discretized equations. A numerically rigorous solution can be obtained by solving the overall discrete eigenvalue system. Mathematical formulations for the EDQ-based DQEM vibration analysis of nonprismatic structures considering the effect of warping torsion are carried out. By using this DQEM model, accurate results of frame problems can efficiently be obtained.

Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

1996 ◽  
Vol 11 (4) ◽  
pp. 371-380 ◽  
Author(s):  
Alphose Zingoni
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Symmetry Group ◽  
Displacement Field ◽  
Three Dimensional ◽  
Beam Elements ◽  
Symmetry Properties ◽  
Mass Matrices ◽  
General Displacement

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

scholarly journals Local Elastic and Geometric Stiffness Matrices for the Shell Element Applied in cFEM

2017 ◽  
Author(s):  
Dávid Visy ◽  
Sándor Ádány
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Shell Element ◽  
Numerical Examples ◽  
Geometric Stiffness ◽  
Stiffness Matrices ◽  
Plane Element ◽  
Unusual Combination ◽  
Shell Finite Element ◽  
Lagrange Strain

In this paper local elastic and geometric stiffness matrices of ashell finite element are presented and discussed. The shell finiteelement is a rectangular plane element, specifically designedfor the so-called constrained finite element method. One of themost notable features of the proposed shell finite element isthat two perpendicular (in-plane) directions are distinguished,which is resulted in an unusual combination of otherwise classicshape functions. An important speciality of the derived stiffnessmatrices is that various options are considered, whichallows the user to decide how to consider the through-thicknessstress-strain distributions, as well as which second-order strainterms to consider from the Green-Lagrange strain matrix. Thederivations of the stiffness matrices are briefly summarizedthen numerical examples are provided. The numerical examplesillustrate the effect of the various options, as well as theyare used to prove the correctness of the proposed shell elementand of the completed derivations.

A new simplified method for anti-symmetric mode in-plane vibrations of frame structures with column cracks

2016 ◽  
Vol 24 (24) ◽  
pp. 5794-5810 ◽  
Author(s):  
Kemal Mazanoglu ◽  
Elif C Kandemir-Mazanoglu
Keyword(s):  
Beam Theory ◽  
Equivalent Model ◽  
Frame Structures ◽  
Symmetric Mode ◽  
Lumped Mass ◽  
Beam Elements ◽  
Local Flexibility ◽  
Mode Vibration ◽  
Frame Systems ◽  
Lumped Masses

This paper is on the natural frequency and mode shape computation of frame structures with column cracks. First, a model of intact frame structures is built to perform vibration analysis. Beam elements are considered as lumped masses and rotational springs at the storey levels of frames. Equivalent model of columns and lumped mass-stiffness effects of beams have been combined to carry out continuous solution for the anti-symmetric mode in-plane vibrations of frames. In addition, frame systems with multiple column cracks are analyzed in terms of anti-symmetric mode vibration characteristics. Cracks are considered as massless rotational springs in compliance with the local flexibility model. Compatibility and continuity conditions are satisfied at crack and storey locations of the equivalent column, modeled using the Euler–Bernoulli beam theory. The proposed method is tested for single-storey single- and multi-bay, H-type and double-storey single-bay frame systems with intact and cracked columns. Results are validated by those given in the current literature and/or obtained by the finite element analyses.

Isogeometric Free Vibration Analysis of Curved Euler–Bernoulli Beams With Particular Emphasis on Accuracy Study

2020 ◽  
pp. 2150011
Author(s):  
Zhuangjing Sun ◽  
Dongdong Wang ◽  
Xiwei Li
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Convergence Rates ◽  
Harmonic Wave ◽  
Free Vibration Analysis ◽  
Curved Beam ◽  
Curved Beams ◽  
Stiffness Matrices ◽  
Accuracy Measure ◽  
Accuracy Study

An isogeometric free vibration analysis is presented for curved Euler–Bernoulli beams, where the theoretical study of frequency accuracy is particularly emphasized. Firstly, the isogeometric formulation for general curved Euler–Bernoulli beams is elaborated, which fully takes the advantages of geometry exactness and basis function smoothness provided by isogeometric analysis. Subsequently, in order to enable an analytical frequency accuracy study, the general curved beam formulation is particularized to the circular arch problem with constant radius. Under this circumstance, explicit mass and stiffness matrices are derived for quadratic and cubic isogeometric formulations. Accordingly, the coupled stencil equations associated with the axial and deflectional displacements of circular arches are established. By further invoking the harmonic wave assumption, a frequency accuracy measure is rationally attained for isogeometric free analysis of curved Euler–Bernoulli beams, which theoretically reveals that the isogeometric curved beam formulation with [Formula: see text]th degree basis functions is [Formula: see text]th order accurate regarding the frequency computation. Numerical results well confirm the proposed theoretical convergence rates for both circular arches and general curved beams.

FREE VIBRATION ANALYSIS OF NON-UNIFORM TIMOSHENKO BEAMS USING DIFFERENTIAL TRANSFORM

1998 ◽  
Vol 22 (3) ◽  
pp. 231-250 ◽  
Author(s):  
Cha’o Kuang Chen ◽  
Shing Huei Ho
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Present Method ◽  
Series Solution ◽  
Free Vibration Analysis ◽  
Timoshenko Beams ◽  
Differential Transform ◽  
Vibration Problems ◽  
Algebraic Operations

This study introduces using differential transform to solve the free vibration problems of a general elastically end restrained non-uniform Timoshenko beam. First, differential transform is briefly introduced. Second, taking differential transform of a non-uniform Timoshenko beam vibration problem, a set of difference equations is derived. Doing some simple algebraic operations on these equations, we can determine any i-th natural frequency, the closed form series solution of any i-th normalized mode shape. Finally, three examples are given to illustrate the accuracy and efficiency of the present method.

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