Vibration of FG porous three-layered beams equipped by agglomerated nanocomposite patches resting on Vlasov's foundation

Equipped sandwich beams (ESBs) are one of the highly demanded structures by the different industries due to their high stiffness to weight ratio. In the present study, the vibrational behavior of a novel ESB is evaluated, analytically. The whole ESB is composed of three layers including functionally graded porous core (FGPC) and two same agglomerated carbon nanofiller reinforced composite (ACNFRC) face sheets. Both nanocomposite layers are constituted from poly(methyl methacrylate) (PMMA) as matrix and CNFs which serve as reinforcing phase. In addition, the effect of agglomeration is considered in nanocomposites, and its tremendous influence on the normalized frequency is depicted in figure and table formats. For the sake of layers properties estimation, power-law and Eshelby–Mori–Tanaka's (EMT)'s approach are hired for, respectively, FGPC and ACNFRCs. Besides this, Hamilton's principle in conjunction with multi-displacement fields and Fourier series analytical method are cooperated tightly to derive motion equations and solve them mathematically. The evaluation of the impacts of various variables as, different displacement fields, thermal environment, CNFs agglomeration, and Vlasov's substrate parameters can be considered as the novelties of this paper. It is revealed that in the context of agglomeration, a higher number of clusters with a lower volume fraction of CNFs inside them can provide higher magnitudes of normalized frequency and consequently rigidity. This work can be assumed as a reference for further future examinations in such a broad context.