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The free in-plane vibration of a shallow circular arch with uniform cross-section is investigated by taking into account axial extension, shear deformation and rotatory inertia effects. The exact solution of the governing differential equations is obtained by the initial value method. By employing the same solution procedure, the solutions are also given for the other cases, in which each effect is considered alone, as well as no effect. The frequency coefficients are obtained for the lowest five vibration modes of arches with five combinations of classical boundary conditions, and various slenderness ratios and opening angles. The results show that the shear deformation and rotatory inertia effects are also very important as well as the axial extension effect, even if a slender shallow arch is considered. The discrepancies among the results of the five cases decrease, when opening angle increases for a constant radius and slenderness ratio. The effects of the boundary conditions and the slenderness ratio of the arch are investigated. The discrepancies among the results of the cases become much more important in higher modes. The mode shapes of a shallow arch are obtained for each case.

Modal analysis of rotating tapered cantilevered Timoshenko beams undergoing in-plane vibration is investigated. The coupling effect of axial motion and transverse motion is considered. The Kane dynamic method is applied to deriving the governing eigenvalue equations. The displacement and rotational angle components are approximately described by the products of Chebyshev polynomials and corresponding boundary functions. Chebyshev polynomials guarantee the numerical robustness while the boundary functions guarantee the satisfaction of the geometric boundary conditions. The excellent convergence of the present solution is exhibited. The results are compared with those available in literature, good agreement is observed. The parametric studies on modal characteristics are presented in detail. The tuned rotational speed is examined and the eigenvalue loci veering phenomenon along with the corresponding mode shapes is investigated.

A new two-dimensional periodic structure is proposed. This structure consists of curved beams with different radii of curvature and lengths spirally connected to each other to form a circle. Geometrical parameter effects on the first three in-plane vibration band gaps of this structure are studied using the differential quadrature method. Results show that for each set of the radii of curvature, as the lengths of inner beam elements increase, all the bands become close to each other. These close band gaps can be considerably enlarged by increasing the difference between the radii of curvature. Having close and wide band gaps means that this structure absorbs in-plane vibrations over a very wide frequency range. The dimensions of this structure are much smaller than other periodic structures due to its unique shape. All the mentioned features make this periodic structure an efficient vibration absorber. Validation of the analytical results is provided through the forced vibration analysis via ANSYS.

The main purpose of this study is to investigate the free vibrations of curved beams, both theoretically and experimentally. Three different geometries for the curved beams are considered in the experiments; circular beam with uniform cross-section, parabolic beams with uniform and varying cross-section. The exact solution of in-plane vibration can be given for only circular beam with uniform cross-section. For all geometries, the finite element solutions were performed by using ANSYS as a pre and post processor to analyze the frequencies and the mode shapes for five different boundary conditions. The results of exact and finite element solutions were then compared with the experimental results for all beams and the comparisons were given in tables and diagrams. Good agreement is obtained between the results.