Dynamic Analysis of a Curved Beam With Tuning of Elastic Modulus and Mass Density in Circumferential Direction

Deformation and stress fields in a curved beam can be tailored by changing its mechanical properties such as the elastic modulus/mass density, which is typically done using functionally graded materials (FGM). Such functional gradation can be done, for instance, by using particles or fiber-reinforced materials with different volume fractions along the beam length. This article presents in-plane vibrations of functionally graded (FG) cantilevered curved beams. Both semi-analytical and finite element modeling are employed to find natural frequencies and mode shapes of such beams. The natural frequencies obtained from the analytical solution and finite element analysis are in close agreement with an error of 6.2% when the variance of material properties gradation is relatively small. In the analytical approach, the direct method is employed to derive the governing linear differential equations of motion. The natural frequencies and mode shapes are obtained using the Galerkin and the finite element methods. First, three natural frequencies and corresponding mode shapes are analyzed for different elastic modulus/mass density distribution functions. Furthermore, the natural frequencies of FG curved beams with a crack are also investigated. Our results indicate that larger cracks near the clamped side of the beam significantly decrease the first natural frequency. In the second and third vibration modes, cracks located in the area with a maximum moment result in the lowest natural frequency values. However, the second and third natural frequencies of the cracked curved beam are not affected by the presence of a crack, if the crack is located at the nodal points of the curved beam.