Dynamic behavior of an electromagnetic nanobeam using the Haar wavelet method and the higher-order Haar wavelet method

The primary objective of this article is to study the vibration characteristics of an electromagnetic nanobeam under the mutual framework of Euler-Bernoulli beam theory and Eringen's nonlocal theory. The nanobeam is assumed to be placed in an electromagnetic field, and the electromagnetic force experienced by the nanobeam is modeled in the present investigation by using Hamilton's principle. The impact of the small-scale parameter, as well as of the Hartmann parameter, is analyzed on the frequency parameter for Hinged-Hinged (H-H), Clamped-Hinged (C-H), Clamped-Clamped (C-C), and Clamped-Free (C-F) edges. Mode shapes are also plotted to exhibit the sensitivity of the Hartmann parameter. Numerical solutions of this model are explored by using two relatively new methods viz. the Haar wavelet method (HWM) and the higher-order Haar wavelet method (HOHWM). Convergence of the present model is analyzed by both the methods and the rate of convergence of both HWM and HOHWM is computed by Richardson's formula. A comparative study is carried out by taking the Hinged-Hinged (H-H) boundary condition as a test case to demonstrate the supremacy of HOHWM over HWM. In order to verify the exactness of the model, the results obtained by the present investigation are compared with other previously published literature in special cases showing admirable agreement.