Analysis of the First Modal Shape Using Two Case Studies

2019 ◽  
Vol 16 (06) ◽  
pp. 1840019 ◽  
Author(s):  
Alexandre de Macêdo Wahrhaftig
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Case Studies ◽  
Stiffness Matrix ◽  
Mass Matrix ◽  
Vibration Modes ◽  
The Finite Element Method ◽  
Modal Shape ◽  
Generalized Eigenvalue ◽  
Eigenvector Analysis

Eigenvector analysis can be performed to determine the shapes of the undamped free vibration modes of a system. Eigenvector analysis involves solving the generalized eigenvalue problem, which considers the stiffness and mass matrix of a structure. For a geometric nonlinear study, both parts of the total stiffness matrix are required. As modal analysis depends on the stiffness, the effect of its reduction on the modal shape of vibration of the structure must be determined. Case studies were evaluated using the finite element method, considering and neglecting the geometric portion of the stiffness matrix. Mathematic functions were applied for comparison.

Development of Bifurcation Analysis Method for Rate Dependent Materials

2019 ◽  
Vol 794 ◽  
pp. 220-225
Author(s):  
Daiki Towata ◽  
Yuichi Tadano
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Bifurcation Analysis ◽  
Stiffness Matrix ◽  
Computational Method ◽  
Analysis Method ◽  
The Finite Element Method ◽  
Tangent Stiffness Matrix ◽  
Rate Dependent ◽  
Element Method

In this study, a novel numerical method to analyze the bifurcation problemof a rate dependent material using the finite element method is proposed. The consistent stiffness matrix, which is required for a bifurcation analysis using the finite element method, for a rate dependent material is generally hard to compute, therefore, a computational method to calculate the tangent stiffness matrix based on a numerical differential is introduced so that exact bifurcation analyses for the rate dependent material can be conducted. A numerical example of the proposed method is demonstrated, and the adequacy of the proposed method is discussed.

On the Unsymmetric Eigenproblem for the Buckling of Shells Under Pressure Loading

1983 ◽  
Vol 50 (1) ◽  
pp. 95-100 ◽  
Author(s):  
H. A. Mang ◽  
R. H. Gallagher
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Hydrostatic Pressure ◽  
Stiffness Matrix ◽  
Circular Cylindrical Shell ◽  
The Finite Element Method ◽  
Equilibrium Equations ◽  
Buckling Loads ◽  
Large Rotations ◽  
Element Method

Consideration of the dependence of hydrostatic pressure on the displacements may result in significant changes of calculated buckling loads of thin arches and shells in comparison with loads calculated without consideration of this effect. The finite element method has made it possible to quantify these changes. On the basis of a shell theory of small displacements but moderately large rotations, this paper derives consistent incremental equilibrium equations for tracing, via the finite element method, the load-displacement path for thin shells subjected to nonuniform hydrostatic pressure and establishes the buckling condition from the incremental equilibrium equations. Within the framework of the finite element method, the character of hydrostatic pressure as one of a follower load is represented in the so-called pressure-stiffness matrix. For shells with loaded free edges, this matrix is unsymmetric. The principal objective of the present paper is to demonstrate that symmetrization of the pressure stiffness matrix resulting from linearization of the buckling condition yields buckling loads that are identical to the eigenvalues resulting from first-order perturbation analysis of the unsymmetric eigenproblem. A circular cylindrical shell with a free and a hinged end, subjected to hydrostatic pressure, is used as an example of the admissibility of symmetrizing the pressure stiffness matrix and for assessing its effect.

Study of Elastic–Plastic Fracture Problem Using Finite Element Technique

1984 ◽  
Vol 106 (4) ◽  
pp. 476-482
Author(s):  
F. T. C. Loo
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Graphical Form ◽  
The Finite Element Method ◽  
Finite Element Technique ◽  
Elastic Plastic ◽  
Center Crack ◽  
Initial Stress Method ◽  
Element Technique

Numerical methods for the analysis of the elastic-plastic fracture problem using a special finite element technique are presented. A brief description of some concepts in elastic-plastic fracture mechanics and of the finite element method is followed by the formulation procedure of the stiffness matrix using eight-noded quadrilateral isoparametric elements. After a terse discussion of the initial stress method, the procedure of computation is extended in the analysis by using an incremental load process. The size and the shape of the plastic zone of a center crack specimen is investigated. Results are presented in graphical form.

scholarly journals Determination of the frequencies and forms of free vibrations of pentagonal plates by the finite-element method

2020 ◽  
pp. 61-66
Author(s):  
A. Grigorenko ◽  
M. Borysenko ◽  
O. Boychuk
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibrations ◽  
Vibration Modes ◽  
The Finite Element Method ◽  
Rigid Attachment ◽  
Equivalent Mass ◽  
Practical Convergence ◽  
Element Method

Frequencies and modes of free vibrations of an isotropic thin pentagonal plate of regular shape with various configurations of rigid attachment at the edges are determined using the finite element method (FEM). The results obtained for some pentagonal plates are compared with the results obtained for square plates of an equivalent mass with corresponding boundary conditions. We present the vibration modes of the studied plates and the topology of the vibration modes for some of the considered plates corresponding to the square plates with free edges and rigidly fixed edges. The reliability of the obtained results is ensured by the use of a substantiated mathematical model, the correct formulation of the problem and the practical convergence of the calculated frequencies when using the FEM.

scholarly journals Comparative analysis of finite element formulations at plane loading of an elastic body

2020 ◽  
Vol 16 (2) ◽  
pp. 139-145
Author(s):  
Natalia A. Gureeva ◽  
Anatoly P. Nikolaev ◽  
Vladislav N. Yushkin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elastic Body ◽  
Stiffness Matrix ◽  
Strain State ◽  
Mixed Formulation ◽  
The Finite Element Method ◽  
Displacement Method ◽  
Internal Forces ◽  
The Difference

The aim of the work - comparison of the results of determining the parameters of the stress-strain state of plane-loaded elastic bodies based on the finite element method in the formulation of the displacement method and in the mixed formulation. Methods. Algorithms of the finite element method in various formulations have been developed and applied. Results. In the Cartesian coordinate system, to determine the stress-strain state of an elastic body under plane loading, a finite element of a quadrangular shape is used in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. The approximation of displacements through the nodal unknowns when obtaining the stiffness matrix of the finite element was carried out using the form function, whose elements were adopted Hermite polynomials of the third degree. Upon receipt of the deformation matrix, the displacements and stresses of the internal points of the finite element were approximated through nodal unknowns using bilinear functions. The stiffness matrix of the quadrangular finite element in the formulation of the displacement method is obtained on the basis of a functional based on the difference between the actual workings of external and internal forces under loading of a solid. The matrix of deformation of the finite element was formed on the basis of a mixed functional obtained from the proposed functional by repla-cing the actual work of internal forces with the difference between the total and additional work of internal forces when loading the body. The calculation example shows a significant advantage of using a finite element in a mixed formulation.

Rotordynamics analysis of a quill-shaft coupling-rotor-bearing system

2014 ◽  
Vol 229 (8) ◽  
pp. 1385-1398 ◽  
Author(s):  
S Feng ◽  
HP Geng ◽  
L Yu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Natural Frequency ◽  
Stiffness Matrix ◽  
Equation Of Motion ◽  
Numerical Results ◽  
The Finite Element Method ◽  
Rotor Bearing ◽  
Bearing System ◽  
Element Method

A quill-shaft coupling-rotor-bearing system is modeled and reported in this paper. The system consists of two rotors connected by a quill-shaft coupling in which each rotor is supported by two bearings. The stiffness matrix of the quill-shaft coupling is deduced and the equation of motion of the system is obtained by using the finite element method. Finally, the rotordynamics analysis of the system is conducted. The numerical results show that more frequency veering points occur for the quill-shaft coupling-rotor-bearing system compared with those of single rotor. In addition, the stiffness of the flexural element has significant effects on the first bending natural frequency of the quill shaft when the length of the quill shaft becomes shorter.

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