*Result*: Effect of Variable Size-Dependent Parameters on the Vibration Characteristics of Axially Functionally Graded Electromagnetic Nanobeams.

*Purpose: This study presents a comprehensive mathematical model for examining the free vibration response of axially functionally graded (AFG) nanobeams influenced by electromagnetic fields, with consideration of spatially varying size-dependent properties. Motivated by the increasing use of AFG nanobeams in nanoelectromechanical systems (NEMS), where electromagnetic actuation and material gradation significantly influence structural performance, this work aims to address the limitations of conventional models that employ constant size-dependent parameters. This study introduces nonlocal and material length scale parameters as variables along the length of the nanobeam, filling an important gap in the existing literature. Methods: The governing equations are derived using Euler-Bernoulli beam theory integrated with nonlocal strain gradient theory to effectively incorporate small-scale effects. The spatial variation of size-dependent parameters follows a power-law distribution consistent with the gradation of material properties. The Rayleigh-Ritz method with orthonormal polynomials is implemented to obtain the frequency parameter values and corresponding mode shapes under various boundary conditions. The accuracy and reliability of the proposed model are validated through convergence studies and comparisons with existing benchmark results from the literature. Results: An extensive parametric study is carried out to investigate the influence of slenderness ratio, power-law exponent, nonlocal and material length scale parameters, Hartmann parameter, and boundary conditions on the first four frequency parameters. The refined modeling framework effectively captures the coupled effects of material gradation, electromagnetic fields, and size dependency on the vibrational response. Conclusion: The findings provide deeper insights into the vibrational behavior of AFG electromagnetic nanobeams and can support the development and optimization of next-generation NEMS, including sensors, actuators, and resilient microdevices operating in electromagnetic environments. [ABSTRACT FROM AUTHOR]

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