Elasticity-based state-space analysis of free vibration in functionally graded CNT-reinforced sandwich beams

Lightweight sandwich panels with nanocomposite cores commonly serve as thermal-protection and energy-absorbing members in aerospace and transportation structures, and their design against vibration requires natural frequencies that remain reliable when the core is thick, functionally graded and orthotropic. A state-space formulation of the Complementary Functions Method (CFM) is presented for variable-coefficient orthotropic plane elasticity and is applied to the free vibration of simply supported sandwich beams whose cores are functionally graded carbon nanotube-reinforced composites (FG-CNTRC). The present study has a twofold aim, namely to remove the numerical fragility introduced by customary condensation of the governing pair in graded orthotropic layers, and to establish elasticity-based benchmark frequencies for configurations where the load-bearing core itself, rather than the face sheets, carries the graded nanotube reinforcement. The coupled second-order governing pair of each layer is solved directly as a first-order system whose coefficient matrix divides only by the physical stiffnesses, therefore the near-singular coefficient combinations produced by the customary condensation into a single fourth-order equation never arise. It is further demonstrated that the fundamental solutions of graded orthotropic layers share a dominant exponentially growing component and that the frequency determinant is formed by the small differences between them. The integration must therefore proceed under adaptive error control, and a fixed-step scheme of comparable order is shown to miss the fundamental frequency by several percent. The nanotubes are aligned with the beam axis, their volume fraction follows the UD, FG-Λ, FG-V, or FG-X pattern with equal total reinforcement, and the effective properties of the orthotropic core obey the refined rule of mixtures. The frequencies of the all-isotropic beam reproduce the published elasticity results, a mirror-symmetry identity between the FG-Λ and FG-V spectra of the symmetric beam is verified numerically, and an independent finite element solution converges from above to the reported values for two graded cores, which the present formulation reproduces at a small fraction of the computational cost. The parametric study covers the volume fraction, the distribution pattern, the slenderness ratio and an unsymmetric face arrangement, and shows that the pattern ranking follows the migration of the modal strain energy while the volume fraction remains the dominant design parameter. Since the solution carries 12 unknowns per frequency search whatever the grading function, seven-digit frequencies are obtained in less than one-fifth of the processing time of the finest finite element mesh reported here, which makes the formulation usable both as a benchmark generator for approximate theories and as a design tool for tailoring the nanotube distribution of vibration-critical panels.