Fourier Analysis of Lowest Normal-Mode Functions for Beams and Plates: Implication for Graphene

The mechanical properties of grapheme sheets and tubes can be modeled by a very nonlinear, integral-partial differential equation. For one (effective) space dimension, the solution u(x,t) is generally written as a product of the mode shape, $\phi $(x), and a mode amplitude, $\phi $(t), i.e., u(x,t)$=\phi $(x)$\psi $(t), where $\phi $(x) is taken to be $\phi $(x)$=$sin($\pi $x/L). This ansatz allows the determination of an ordinary differential equation for $\psi $(t), which turns out to be the Duffing equation. However, the (constant) coefficients appearing in the Duffing equation depend on exactly which specific function is selected for $\phi $(x). Since these coefficients are used to calculate various mechanical features, it follows that the use of different $\phi $(x) can produce different estimates for the related mechanical properties. We consider the case of a rigidly clamped beam, defined by the conditions \begin{equation} \phi (0)=\phi (L)=0, \phi' (0)=\phi' (L)=0, \end{equation} and compare its Fourier series representations to the functions \begin{equation} \phi 1(x)=\sin(\pi x/L), \phi \textunderscore \{^2\} \textbraceright (x)=x(L-x). \end{equation} An example of a mode function satisfying Eq.(1) is \begin{equation} \phi 3(x)=x^2(L-x)^2. \end{equation} Background information on these issues is given in I. Kovavic and M. J. Brennan: The Duffing Equation (Wiley, 2011), sections 2.8 -- 2.10.