Exact dynamic stiffness matrix for a thin-walled beam of doubly asymmetric cross-section filled with shear sensitive material

2006 ◽  
Vol 69 (13) ◽  
pp. 2758-2779 ◽  
Author(s):  
B. Rafezy ◽  
W. P. Howson
Keyword(s):  
Cross Section ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Sensitive Material ◽  
Dynamic Stiffness Matrix ◽  
Thin Walled

Free Vibration of Helicoidal Beams of Arbitrary Shape and Variable Cross Section

2002 ◽  
Vol 124 (3) ◽  
pp. 397-409 ◽  
Author(s):  
Wisam Busool ◽  
Moshe Eisenberger
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Rotational Inertia ◽  
Variable Cross ◽  
Dynamic Stiffness Matrix ◽  
Helical Springs

In this study, the dynamic stiffness method is employed for the free vibration analysis of helical springs. This work gives the exact solutions for the natural frequencies of helical beams having arbitrary shapes, such as conical, hyperboloidal, and barrel. Both the cross-section dimensions and the shape of the beam can vary along the axis of the curved member as polynomial expressions. The problem is described by six differential equations. These are second order equations with variable coefficients, with six unknown displacements, three translations, and three rotations at every point along the member. The proposed solution is based on a new finite-element method for deriving the exact dynamic stiffness matrix for the member, including the effects of the axial and the shear deformations and the rotational inertia effects for any desired precision. The natural frequencies are found as the frequencies that cause the determinant of the dynamic stiffness matrix to become zero. Then the mode shape for every natural frequency is found. Examples are given for beams and helical springs with different shape, which can vary along the axis of the member. It is shown that the present numerical results agree well with previously published numerical and experimental results.

scholarly journals Analytical Investigation of the Dynamic Response of a Timoshenko Thin-Walled Beam with Asymmetric Cross Section Under Deterministic Loads

2017 ◽  
Vol 11 (1) ◽  
pp. 802-821
Author(s):  
Elham Ghandi ◽  
Ahmed Ali Akbari Rasa
Keyword(s):  
Differential Equations ◽  
Dynamic Response ◽  
Stiffness Matrix ◽  
Matrix Method ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Dynamic Features ◽  
Dynamic Stiffness Matrix ◽  
Thin Walled

Inroduction: The objective of the present paper is to analyze dynamic response of the Timoshenko thin-walled beam with coupled bending and torsional vibrations under deterministic loads. The governing differential equations were obtained by using Hamilton’s principle. The Timoshenko beam theory was employed and the effects of shear deformations, Rotary inertia and warping stiffness were included in the present formulations. Dynamic features of underlined beam are obtained using free vibration analysis. Methods: For this purpose, the dynamic stiffness matrix method is used. Application of exact dynamic stiffness matrix method on the movement differential equations led to the issue of nonlinear eigenvalue problem that was solved by using Wittrick–Williams algorithm . Differential equations for the displacement response of asymmetric thin-walled Timoshenko beams subjected to deterministic loads are used for extracting orthogonality property of vibrational modes. Results: Finally the numerical results for dynamic response in a sample of mentioned beams is presented. The presented theory is relatively general and can be used for various kinds of deterministic loading in Timoshenko thin-walled beams.

DYNAMIC STIFFNESS MATRIX FOR FLEXURAL-TORSIONAL, LATERAL BUCKLING AND FREE VIBRATION ANALYSES OF MONO-SYMMETRIC THIN-WALLED COMPOSITE BEAMS

2009 ◽  
Vol 09 (03) ◽  
pp. 411-436 ◽  
Author(s):  
NAM-IL KIM ◽  
DONG KU SHIN
Keyword(s):  
Finite Element ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Elastic Strain Energy ◽  
Composite Beams ◽  
Energy Principle ◽  
Dynamic Stiffness Matrix ◽  
Axial Forces ◽  
Thin Walled ◽  
Force Displacement

This paper presents the elastic strain energy, the potential energy with the second order terms of finite rotations, and the kinetic energy with rotary inertia effect for thin-walled composite beams of mono-symmetric cross-section. The equations of motion and force-displacement relationships are derived from the energy principle and explicit expressions for displacement parameters are given based on power series expansions of displacement components. The exact dynamic stiffness matrix is determined using the force-displacement relationships. In addition, the finite element model based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of the formulation, numerical examples are solved and the solutions are compared with results from ABAQUS's shell elements, analytical solutions from previous researchers and the finite element solutions using the Hermitian beam elements. The influence of constant and linearly variable axial forces, fiber orientation, and boundary conditions on the vibration behavior of composite beam are also investigated.

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