Dynamic Stiffness Analysis of a Beam Based on Trigonometric Shear Deformation Theory

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Jun Li ◽  
Hongxing Hua ◽  
Rongying Shen
Keyword(s):  
Shear Deformation ◽  
Stiffness Matrix ◽  
Deformation Theory ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Beam Element ◽  
Mode Shapes ◽  
Dynamic Stiffness Matrix ◽  
Stiffness Analysis

The dynamic stiffness matrix of a uniform isotropic beam element based on trigonometric shear deformation theory is developed in this paper. The theoretical expressions for the dynamic stiffness matrix elements are found directly, in an exact sense, by solving the governing differential equations of motion that describe the deformations of the beam element according to the trigonometric shear deformation theory, which include the sinusoidal variation of the axial displacement over the cross section of the beam. The application of the dynamic stiffness matrix to calculate the natural frequencies and normal mode shapes of two rectangular beams is discussed. The numerical results obtained are compared to the available solutions wherever possible and validate the accuracy and efficiency of the present approach.

EXACT VIBRATION FREQUENCIES AND MODES OF BEAMS WITH INTERNAL RELEASES

2002 ◽  
Vol 02 (01) ◽  
pp. 63-75 ◽  
Author(s):  
M. EISENBERGER
Keyword(s):  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Beam Element ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Dynamic Stiffness Matrix ◽  
Continuous Beams ◽  
Stiffness Method ◽  
Dynamic Stiffness Method

The exact vibration frequencies of continuous beams with internal releases are found using the dynamic stiffness method. Two types of releases are considered: hinge and sliding discontinuities. First, the exact dynamic stiffness matrix for a beam element with a release is derived and then used in the assembly of the structure dynamic stiffness matrix. The natural frequencies are found as the values of frequency that make this matrix singular. Then the mode shapes are found exactly. Examples are given for continuous beams with different releases.

scholarly journals Free vibration characteristics of thick doubly curved variable stiffness composite laminated shells using higher-order shear deformation theory

2018 ◽  
Vol 172 ◽  
pp. 03010 ◽  
Author(s):  
Anand Venkatachari ◽  
K. Ramajeyathilagam
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Variable Stiffness ◽  
Mode Shapes ◽  
Laminated Shells ◽  
Surface Displacements ◽  
Edge Conditions ◽  
Geometric Variables

In the present work, the natural frequencies of cylindrical and spherical laminated shells with variable stiffness are numerically studied using a shear flexible isogeometric finite element. The kinematics relies on cubic shear deformation theory in which cubic variation is assumed for the surface displacements and a quadratic variation for the traverse displacement along the thickness. A zig-zag function, used for the in-plane displacements, accounts for the abrupt discontinuity at the boundaries of the laminae. The Lagrangian equations of motion is deployed to solve the frequencies of curved panels. A detailed parametric analysis examines the influence of fibre centre/edge angles, shell geometric variables, material anisotropy and edge conditions on frequencies and mode shapes.

COUPLED FREE VIBRATION ANALYSIS OF SHEAR-FLEXIBLE LAMINATED COMPOSITE I-BEAMS

2013 ◽  
Vol 21 (01) ◽  
pp. 1250024
Author(s):  
NAM-IL KIM
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Stiffness Matrix ◽  
Vibration Analysis ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Dynamic Stiffness Matrix ◽  
Analytical Technique

The coupled free vibration analysis of the thin-walled laminated composite I-beams with bisymmetric and monosymmetric cross sections considering shear effects is developed. The laminated composite beam takes into account the transverse shear and the restrained warping induced shear deformation based on the first-order shear deformation beam theory. The analytical technique is used to derive the constitutive equations and the equations of motion of the beam in a systematic manner considering all deformations and their mutual couplings. The explicit expressions for displacement parameters are presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the dynamic stiffness matrix is determined using the force–displacement relationships. In addition, for comparison, a finite beam element with two-nodes and fourteen-degrees-of-freedom is presented to solve the equations of motion. The performance of the dynamic stiffness matrix developed by study is tested through the solutions of numerical examples and the obtained results are compared with results available in literature and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The vibrational behavior and the effect of shear deformation are investigated with respect to the modulus ratios and the fiber angle change.

Finite element bending analysis of symmetric and non-symmetric functionally graded sandwich beams using a novel parabolic shear deformation theory

2021 ◽  
pp. 146442072110050
Author(s):  
Mohamed-Ouejdi Belarbi ◽  
Abdelhak Khechai ◽  
Aicha Bessaim ◽  
Mohammed-Sid-Ahmed Houari ◽  
Aman Garg ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Element Model ◽  
Beam Element ◽  
Functionally Graded ◽  
Sandwich Beams

In this paper, the bending behavior of functionally graded single-layered, symmetric and non-symmetric sandwich beams is investigated according to a new higher order shear deformation theory. Based on this theory, a novel parabolic shear deformation function is developed and applied to investigate the bending response of sandwich beams with homogeneous hardcore and softcore. The present theory provides an accurate parabolic distribution of transverse shear stress across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the functionally graded sandwich beam without using any shear correction factors. The governing equations derived herein are solved by employing the finite element method using a two-node beam element, developed for this purpose. The material properties of functionally graded sandwich beams are graded through the thickness according to the power-law distribution. The predictive capability of the proposed finite element model is demonstrated through illustrative examples. Four types of beam support, i.e. simply-simply, clamped-free, clamped–clamped, and clamped-simply, are used to study how the beam deflection and both axial and transverse shear stresses are affected by the variation of volume fraction index and beam length-to-height ratio. Results of the numerical analysis have been reported and compared with those available in the open literature to evaluate the accuracy and robustness of the proposed finite element model. The comparisons with other higher order shear deformation theories verify that the proposed beam element is accurate, presents fast rate of convergence to the reference results and it is also valid for both thin and thick functionally graded sandwich beams. Further, some new results are reported in the current study, which will serve as a benchmark for future research.

A New Finding on the Dynamic Stiffness Matrices of Asymmetric and Axisymmetric Shafts

2001 ◽  
Author(s):  
Francesco A. Raffa ◽  
Furio Vatta
Keyword(s):  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Rotational Inertia ◽  
Dynamic Stiffness Matrix ◽  
Rayleigh Beam ◽  
Principal Moments Of Inertia ◽  
Stiffness Matrices ◽  
New Finding

Abstract In this paper the dynamic stiffness method is developed to analyze a rotating asymmetric shaft, i.e. a shaft whose transverse section is characterized by dissimilar principal moments of inertia. The shaft is modeled according to the Rayleigh beam theory including the effects of both translational and rotational inertia, and gyroscopic moments. The mathematical description is carried out in a reference system rotating at the shaft speed and is based on the exact solution of the governing differential equations of motion. The exact expressions of the shaft displacements are utilized for deriving the 8 × 8 complex dynamic stiffness matrix of the shaft. A new relationship is obtained which links the dynamic stiffness matrix of the asymmetric shaft to the 4 × 4 real dynamic stiffness matrix of the axisymmetric shaft.

Vibration control of magnetostrictive plate under multi- physical loads via trigonometric higher order shear deformation theory

2015 ◽  
Vol 23 (19) ◽  
pp. 3057-3070 ◽  
Author(s):  
Ali Ghorbanpour Arani ◽  
Z Khoddami Maraghi ◽  
H Khani Arani
Keyword(s):  
Elastic Medium ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Feedback Gain ◽  
First Time ◽  
Free Vibration Response

For the first time in this research, a feedback control system is used to study the free vibration response of rectangular plate made of magnetostrictive material. In this regard, magnetostrictive plate (MsP) is analyzed by trigonometric higher order shear deformation theory that involved six unknown displacement functions and does not require shear correction factor. The MsP is supported by elastic medium as Pasternak foundation which considers both normal and shears modules. Also the MsP undergoes in-plane forces in x and y directions. Considering simply supported boundary condition, six equations of motion are derived using Hamilton’s principle and solved by differential quadrature method. Results indicate the effect of aspect ratio, thickness ratio, elastic medium, compression and tension loads on vibration behavior of MsP. Also, findings show the controller effect of velocity feedback gain to minimize the frequency as far as other parameters become ineffective. These findings can be used to active noise and vibration cancellation systems in many structures.

Sign in / Sign up

Export Citation Format

Share Document