Dynamical behavior of nanobeam embedded in constant, linear, parabolic, and sinusoidal types of Winkler elastic foundation using first-Order nonlocal strain gradient model

In this research, the differential quadrature method is employed to investigate the nonlocal vibration of nanobeam resting on various types of Winkler elastic foundations such as constant, linear, parabolic, and sinusoidal types. The nanobeam is modeled with Winkler elastic foundation considering the elastic coefficient varying along the axis of the nanobeam. Within the framework of Euler-Bernoulli beam theory, first order strain gradient model is incorporated to compute the frequency parameters for Hinged-Hinged (H-H) and Clamped-Hinged (C-H) boundary conditions. A convergence study is also performed to demonstrate the efficiency, adequacy, and reliability of the method. Further, the results are compared with available data of previously published research in special cases showing robust agreement. Likewise, the effects of the nonlocal parameter, strain gradient parameter, non-uniform parameters, and Winkler modulus parameter on the frequency parameters are studied comprehensively.