Vibration Analysis of Nonlinear Tapered Functionally Graded Beams Using
Collocation Method with Gauss-Legendre Points
- Reza Adelkhani,
- Jaafar Ghanbari
Abstract
Free transverse vibration of variable cross-section cantilever FG beams
with nonlinear profiles is investigated in this paper. Four thickness
functions, namely, linear, parabolic, sinusoidal, and exponential
functions are assumed for variation of the cross-section of the beam.
Linear and exponential grading rules are covered in this paper. The
governing differential equation is solved using the weighted residual
collocation method with the exact solution shape functions for the
uniform beam as the trial functions. This choice for the trial functions
showed an increase in the convergence rate. The Gauss-Legendre points
are used as the collocation points to reduce the fluctuations in the
convergence curve as usually encountered in the point collocation
method. The effects of the taper parameter for all kinds of thickness
functions on the natural frequencies are studied. The effects of various
parameters, including taper parameter, profile, and grading function on
the natural frequencies are investigated. Also, a series of finite
element simulations are performed for comparison purposes. It is
observed that the obtained results are in good agreement with the
numerical simulations and available published data with vastly reduced
computational cost as a result of using the collocation method.