This papers deals with nonlinear vibrations of non-uniform beams with geometrical nonlinearities such as moderately large curvatures, and inertia nonlinearities such as longitudinal and rotary inertia forces. The nonlinear fourth-order partial-differential equation describing the above nonlinear effects is presented. Using the method of multiple scales, each effect is found by reducing the nonlinear partial-differential equation of motion to two simpler linear partial-differential equations, homogeneous and nonhomogeneous. These equations along with given boundary conditions are analytically solved obtaining so-called zero-and first-order approximations of the beam’s nonlinear frequencies. Since the effect of mid-plane stretching is ignored, any boundary conditions could be considered as long as the supports are not fixed a constant distance apart. Analytical expressions showing the influence of these three nonlinearities on beam’s frequencies are presented up to some constant coefficients. These coefficients depend on the geometry of the beam. This paper can be used to study these influences on frequencies of different classes of beams. However, numerical results are presented for uniform beams. These results show that as beam slenderness increases the effect of these nonlinearities decreases. Also, they show that the most important nonlinear effect is due to moderately large curvature for slender beams.
The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order