Analysis of the dynamic stability of a functional graded Euler-Bernoulli beam and variable crosssection using the Ritz method

In this article, the dynamic stability of a functionally graded Euler-Bernoulli beam with a variable cross-section under dynamic axial load is investigated. The variable cross-section is assumed to be an exponential function and the material is a combination of aluminum and subconium oxide. In the first step, the governing differential equation is derived using Hamilton's method. In the next step, the weak form of the equation is calculated and the Chebyshev series is used as the transverse displacement function and the Bolotin function is used as the time series. Then, the material, geometric, and mass hardness matrices are extracted. Finally, the values of the dynamic load factor are determined for different excitation frequencies. The increase in the coefficient of changes in the level and moment of inertia, the dimensionless static load coefficient and the power of the functional material causes the stiffness of the beam to decrease and causes the changes in the dimensionless dynamic load coefficient to be transferred to smaller frequencies.