Forward and inverse solutions for hygro-magneto vibration of Euler nanobeam in thermal environment

The objectives of this article are two fold—first analyze the vibration behavior of Euler nanobeam under hygro-magnetic-thermal environment resting on a Winkler–Pasternak foundation as a forward process, based on two semi analytical techniques: Differential Quadrature Method (DQM) and Differential Transform Method (DTM), second introduced DTM-based inverse problem to obtain the system parameters by using the obtained frequency parameters by forward problem. The governing differential equation is obtained by Hamilton’s principle and the nonlocal strain gradient theory is implemented to capture the nanoscale effects. A complete mathematical process for DQM and DTM has been discussed and vibration frequencies are obtained by using both methods under three different classical boundary conditions: Simply Supported–Simply Supported (SS), Clamped–Simply Supported (CS), and Clamped–Clamped (CC) in the forward problem. In the inverse case, those obtained frequencies are used to find the unknown parameters by inverse DTM. A convergence study for both forward and inverse methods is discussed in terms of frequency and other characteristic parameters under these three boundary conditions. Also the effects of nonlocal parameters, length scale parameters, Winkler–Pasternak foundation, Hygro-Magnetic and Thermal parameters on the vibration have been discussed by tables and graphical results. The main novelty of this work is that DTM-based inverse problem is introduced, which can be extended in experimental works in future research.