This paper presents a method for a free vibration analysis of a thin-walled beam of doubly asymmetric cross section filled with shear sensitive material. In the study, first of all, a dynamic transfer matrix method was obtained for planar shear flexure and torsional motion. Then, uncoupled angular frequencies were obtained by using dynamic element transfer matrices and boundary conditions. Coupled frequencies were obtained by the well-known two-dimensional approaches. At the end of the study, a sample taken from the literature was solved, and the results were evaluated in order to test the convenience of the method.
In the last two decades research on the dynamics of beams has grown enormously. There are numerous studies [
A dynamic transfer matrix method for the free vibration analysis of a thin-walled beam of doubly asymmetric cross-section filled with shear sensitive materials is suggested in this study. The following assumptions are made in this study: the behaviour of the material is linear elastic, small displacement theory is valid, and the dynamic coupling effect of structure caused by the eccentricity between the center of shear rigidity and the flexural rigidity center is ignored in analysis.
Figure
Typical thin-walled beam [
The governing equations for
If a sinusoidal variation of
Substituting (
For the Clamped-Free: Clamped-Clamped: Simply-Simply: Free-Free: Clamped-Simply:
In frequency equations the values of
Ignoring the dynamic coupling effect of structure caused by the eccentricity between the center of shear rigidity and the geometric center the coupled frequencies of the shear torsional beam can be obtained by using uncoupled frequencies and the well-known equation as follows [
A program that considers the method presented in this study as a basis has been prepared in MATLAB, and the operation stages are presented below. element dynamic Transfer matrices are calculated for each element by using ( System dynamic transfer matrices (see ( The angular frequencies of uncoupled vibrations are obtained by using the boundary conditions. The coupled angular frequencies are found by using (
In this part of the study two numerical examples were solved by a program written in MATLAB to validate the presented method. The results are compared with those given in the literature.
The first example considers the beam studied by Tanaka and Bercin [
The first three coupled natural frequencies of the beam are calculated by the presented method and compared with the results by Tanaka and Bercin [
Coupled natural frequencies for the beam of example 1.
Natural frequencies (Hz) | |||||||||
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BC | Proposed method |
Tanaka and Bercin [ |
Rafezy and Howson [ | ||||||
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| |
C-F | 17.17 | 27.31 | 59.10 | 17.03 | 27.58 | 59.25 | 17.17 | 27.31 | 59.10 |
S-S | 44.71 | 75.14 | 164.87 | 41.48 | 74.12 | 164.11 | 44.71 | 75.14 | 164.87 |
A typical continuous beam with a doubly asymmetric cross section is considered in this example (Figure
The doubly asymmetric, continuous channel section and the cross section of beam of example 2 with warping allowed at B, C, and D but fully constrained at A.
The beam comprises a thin-walled outer layer and a shear core with the following properties between support points A and B. The typical uniform thin-walled beam has a length of 1.5 m with a doubly asymmetric cross section. The properties of the cross section are as follows:
The shear core is omitted between points B and D, where the cross-sectional properties remain unchanged, except that
The first three coupled natural frequencies of the beam are calculated by the presented method and compared with the results of Rafezy and Howson [
Coupled natural frequencies of the continuous beam of example 2.
Frequency number | This study | Rafezy and Howson[ |
Difference (%) |
---|---|---|---|
1 | 6.906 | 6.940 | −0.49 |
2 | 19.763 | 19.796 | −0.17 |
3 | 35.461 | 33.836 | 4.80 |
The main source of error between the proposed method and Rafezy and Howson methods is the eccentricity between the center of shear stiffness and flexural stiffness which was not taken into account in the proposed method.
This paper presents a method for a free vibration analysis of a thin-walled beam of doubly asymmetric cross section filled with shear sensitive material. In the study, first of all, a dynamic transfer matrix method was obtained for planar shear flexure and torsional motion. Then, uncoupled angular frequencies were obtained by using dynamic element transfer matrices and boundary conditions. Coupled frequencies were obtained by the well-known two-dimensional approaches. It was observed from the sample taken from the literature that the presented method gave sufficient results. The error margin of the proposed method is shown to be less than 5%. The main source of error is the eccentricity between the center of shear stiffness and flexural stiffness which was not taken into account in the proposed method.
The transfer matrix method is an efficient and computerized method which also provides a fast and practical solution since the dimension of the matrix for the elements and system never changes. Because of this the proposed method is simple and accurate enough to be used both at the concept design stage and for final analyses.