This study examines the nonlinear vibration behavior of truncated conical micro-shells reinforced with TiO₂ nanoparticles, with particular emphasis on the role of localized nanoparticle agglomeration. Size-dependent governing equations are derived by combining Donnell shell theory and nonlinear von Kármán kinematics within the framework of nonlocal strain-gradient elasticity. Applying the Galerkin method yields the characteristic equation and a set of coupled nonlinear differential equations. The nonuniform distribution and agglomeration of nanoparticles are incorporated through a functionally graded representation of the particle content together with an extended rule of mixtures. A comprehensive parametric analysis demonstrates that nanoparticle agglomeration markedly reduces the natural frequency of the structure. For typical ranges of the agglomeration parameters, the fundamental natural frequency decreases by approximately 44–74%, and may reach reductions of up to about 90% in resonance conditions. It is also shown that increasing the nanoparticle volume fraction from 0.1 wt% to 0.5 wt% under agglomerated dispersion does not proportionally enhance stiffness; the associated improvement in natural frequency remains marginal, whereas agglomeration-induced softening dominates the response. Furthermore, increasing the agglomeration intensity can reduce the maximum steady-state vibration amplitude by about 60% due to energy localization. Overall, the results indicate that uniform nanoparticle dispersion yields the highest natural frequencies, while pronounced agglomeration significantly degrades dynamic performance. Within the investigated parameter range, lower TiO₂ contents may be preferable when additional nanoparticles promote clustering more strongly than stiffness enhancement; however, this recommendation should be used together with experimentally measured dispersion quality and the actual manufacturing route.
Salmani et al. (Fri,) studied this question.