Transverse Natural Vibrations of a Cracked Beam Loaded with a Constant Axial Force

1993 ◽  
Vol 115 (4) ◽  
pp. 524-528 ◽  
Author(s):  
M. Krawczuk ◽  
W. M. Ostachowicz
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Numerical Calculations ◽  
Open Crack ◽  
Natural Vibrations ◽  
Cracked Beam ◽  
Matrix Calculation

The influence of transverse, one-edge open cracks on the natural frequencies of the cantilever beam subjected to vertical loads is analyzed. A finite element method (FE) is used for modelling the beam. A part of the cracked beam is modelled by beam finite elements with an open crack. Parts of the beam without a crack are modelled by standard beam finite elements. An algorithm of a linear stiffness matrix and a geometrical stiffness matrix calculation for a cracked element is presented. The results of numerical calculations obtained for the presented model are compared with the results of analytical calculations given in the literature and also with the results of numerical calculations obtained for a model with geometrical stiffness matrix of uncracked elements.

scholarly journals Fast Stiffness Matrix Calculation for Nonlinear Finite Element Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Emir Gülümser ◽  
Uğur Güdükbay ◽  
Sinan Filiz
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Nonlinear Finite Element ◽  
Nonlinear Finite Element Method ◽  
Nonlinear Fem ◽  
Calculation Technique ◽  
Stiffness Matrices ◽  
Matrix Calculation ◽  
Element Method

We propose a fast stiffness matrix calculation technique for nonlinear finite element method (FEM). Nonlinear stiffness matrices are constructed using Green-Lagrange strains, which are derived from infinitesimal strains by adding the nonlinear terms discarded from small deformations. We implemented a linear and a nonlinear finite element method with the same material properties to examine the differences between them. We verified our nonlinear formulation with different applications and achieved considerable speedups in solving the system of equations using our nonlinear FEM compared to a state-of-the-art nonlinear FEM.

Natural Vibrations of a Clamped-Clamped Arch With an Open Transverse Crack

1997 ◽  
Vol 119 (2) ◽  
pp. 145-151 ◽  
Author(s):  
M. Krawczuk ◽  
W. Ostachowicz
Keyword(s):  
Finite Element ◽  
Stiffness Matrix ◽  
Edge Crack ◽  
Natural Frequencies ◽  
Element Model ◽  
Mode Shapes ◽  
Curved Beam ◽  
Natural Vibrations ◽  
Crack Location ◽  
Cracked Arch

The paper presents a finite element model of the arch with a transverse, one-edge crack. A part of the cracked arch is modelled by a curved beam finite element with the crack. Parts of the arch without the crack are modelled by noncracked curved beam finite elements. The crack occurring in the arch is nonpropagating and open. It is assumed that the crack changes only the stiffness of the arch, whereas the mass is unchanged. The method of the formation of the stiffness matrix of a curved beam finite element with the crack is presented. The effects of the crack location and its length on the changes of the in-plane natural frequencies and mode shapes of the clamped-clamped arch are studied.

Derivation of New Transcendental Member Stiffness Determinant for Vibrating Frames

2003 ◽  
Vol 03 (02) ◽  
pp. 299-305 ◽  
Author(s):  
F. W. Williams ◽  
D. Kennedy
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Infinite Order ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Structural Stiffness ◽  
Trial And Error ◽  
Stiffness Matrices ◽  
Vibration Problems

Transcendental dynamic member stiffness matrices for vibration problems arise from solving the governing differential equations to avoid the conventional finite element method (FEM) discretization errors. Assembling them into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or their squares) are found with certainty using the Wittrick–Williams algorithm. This paper gives equations for the recently discovered transcendental member stiffness determinant, which equals the appropriately normalized FEM dynamic stiffness matrix determinant of a clamped ended member modelled by infinitely many elements. Multiplying the overall transcendental stiffness matrix determinant by the member stiffness determinants removes its poles to improve curve following eigensolution methods. The present paper gives the first ever derivation of the Bernoulli–Euler member stiffness determinant, which was previously found by trial-and-error and then verified. The derivation uses the total equivalence of the transcendental formulation and an infinite order FEM formulation, which incidentally gives insights into conventional FEM results.

scholarly journals Rayleigh's quotient for multiple cracked beam and application

2011 ◽  
Vol 33 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Nguyen Tien Khiem ◽  
Tran Thanh Hai
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Analysis ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Shape Function ◽  
Detection Problem ◽  
Cracked Beam ◽  
Rayleigh’S Quotient ◽  
Good Agreement

Rayleigh's quotient for Euler-Bernoulli multiple cracked beam with different boundary conditions has been derived from the governed equation of free vibration. An appropriate choosing of approximate shape function in terms of mode shape of uncracked beam and specific functions satisfying conditions at cracks and boundaries leads to an explicit expression of natural frequencies through crack parameters that can simplify not only the analysis of natural frequencies of cracked beam but also the crack detection problem. Numerical analysis of natural frequencies of the cracked beam by using the obtained expression in comparison with the well-known methods such as the characteristic equation and finite element method shows their good agreement. The analytical expression of natural frequencies applied to the crack detection problem allows the result of detection to be improved.

scholarly journals Orthogonality of Modes of Structures When Using the Exact Transcendental Stiffness Matrix Method

2000 ◽  
Vol 7 (1) ◽  
pp. 23-28 ◽  
Author(s):  
K.L. Chan ◽  
F.W. Williams
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Stiffness Matrix ◽  
Matrix Method ◽  
Natural Frequencies ◽  
Plane Frames ◽  
Transcendental Functions ◽  
Physical Insight ◽  
Piecewise Continuous

This paper presents theory, physical insight and results for mode orthogonality of piecewise continuous structures, including both coincident and non-coincident natural frequencies. The structures are ones for which exact member equations have been obtained by solving the governing differential equations, e.g. as can be done for members of plane frames or prismatic plate assemblies. Such member equations are transcendental functions of the distributed member mass and the frequency. They are used to obtain a transcendental overall stiffness matrix for the structure, from which the natural frequencies are extracted by using the Wittrick-Williams algorithm, prior to using any existing method to find the modes which are examined from the orthogonality viewpoint in this paper. The natural frequencies and modes found are the exact values for the structure in the sense that the usual finite element method approximations are avoided.

Vibration Analysis of a Timoshenko Beam with Transverse Open Crack by Finite Element Method

2014 ◽  
Vol 592-594 ◽  
pp. 2102-2106 ◽  
Author(s):  
Alok Ranjan Biswal ◽  
Rabindra Kumar Behera ◽  
Tarapada Roy
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Minimum Weight ◽  
Structural Defects ◽  
Mode Shapes ◽  
Open Crack ◽  
Engineering Structures ◽  
Element Method

The design of structures and machineries in present days are based on optimizing of multi-objectives such as maximum strength, maximum life, minimum weight and minimum cost. Due to this flexiblity they allow having a very high level of stresses. This leads to development of cracks in their elements. Due to long-term service many engineering structures may have structural defects such as cracks. So it is very much essential to know the property of structures and response of such structures in various cases. In this article the natural frequencies and mode shapes of an uncracked and cracked cantilever Timoshenko beam is studied by using finite element method (FEM) and MATLAB programme. The effect of crack on the natural frequencies of the uncracked and cracked Timoshenko beam is studied.

scholarly journals Modeling and Eigenfrequency Analysis of Sound-Structure Interaction in a Rectangular Enclosure with Finite Element Method

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Samira Mohamady ◽  
Raja Kamil Raja Ahmad ◽  
Allahyar Montazeri ◽  
Rizal Zahari ◽  
Nawal Aswan Abdul Jalil
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Frequencies ◽  
Active Noise Control ◽  
Interior Noise ◽  
Rectangular Enclosure ◽  
Sound Structure ◽  
Flexible Panel ◽  
Experimental Works ◽  
Element Method

Vibration of structures due to external sound is one of the main causes of interior noise in cavities like automobile, aircraft, and rotorcraft, which disturb the comfort of passengers. Accurate modelling of such phenomena is required in eigenfrequency analysis and in designing an active noise control system to reduce the interior noise. In this paper, the effect of periodic noise travelling into a rectangular enclosure is investigated with finite element method (FEM) using COMSOL Multiphysics software. The periodic acoustic wave is generated by a point source outside the enclosure and propagated through the enclosure wall and excites an aluminium flexible panel clamped onto the enclosure. The behaviour of the transmission of sound into the cavity is investigated by computing the modal characteristics and the natural frequencies of the cavity. The simulation results are compared with previous analytical and experimental works for validation and an acceptable match between them were obtained.

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