تحلیل ارتعاش آزاد محوری میله‌های مدرج تابعی با روش اجزای محدود دینامیکی

Accurate analysis of axial vibrations in bars with spatially varying material properties has always been one of the major challenges in computational mechanics and structural analysis due to the complexity arising from the continuous variation of mechanical properties along the length. In the present study, an approach based on the formulation of the dynamic stiffness matrix is developed, which, by employing trigonometric shape functions, is capable of precisely modeling the dynamic behavior of functionally graded bars subjected to axial vibration. Unlike the classical finite element method, which in most cases uses polynomial shape functions and results in considerable numerical errors—especially in dynamic problems with a small number of elements—this method inherently incorporates dynamic effects into the stiffness matrix and significantly enhances computational accuracy. The axial properties of the functionally graded materials in this study are defined based on power-law functions and are included in the formulation of the mass and stiffness matrices. Subsequently, using modal analysis and iterative algorithms, the eigenvalues—and consequently the natural frequencies—of the problem are extracted with high accuracy. To evaluate the validity and efficiency of the proposed method, the developed numerical model is compared with an analytical benchmark example, yielding excellent agreement. The results indicate that the use of polynomial shape functions in vibration modeling can reduce the accuracy of predicting the dynamic response; therefore, using only one element in the classical finite element method is not recommended for analyzing such problems. Moreover, due to the increased analysis error associated with heterogeneous bars in the classical finite element method, the use of this method for functionally graded materials should be considered with greater caution.