## 1 Introduction
For free vibration analysis, they obtained the decoupled sixth-order differential equation, and for buckling and bending analysis, a fifth-order differential equation was obtained. Finally, a system of homogeneous algebraic equations was derived. The nonlinear vibration of a functionally graded CNT sandwich Timoshenko beam resting on a Pasternak foundation was investigated by [PERSON] and [PERSON] [11]. The material properties were estimated by the Eshelby-Mori-Tanaka approach, while the differential quadrature method was applied to obtain the natural frequency. The elastic vibration and the bending behavior of a novel nanocomposite beam were examined by [PERSON] et al. [12] by improving the third-order shear deformation theory. To describe different boundary conditions, they used the Chebyshev-Ritz method, and the modified Halpin-Tsai micromechanics model was used to evaluate the effective Young's modulus. Critical buckling load and maximum deflection of a nanobeam with the help of [PERSON] and Timoshenko higher-order beam theories were analyzed by [PERSON] et al. [13]. The governing equations were established by NSGT incorporating surface effects. The [PERSON]'s approach was used for simply supported boundary conditions. The Eshelby-Mori-Tanaka approach was investigated by [PERSON] and [PERSON] [14] for aligned and randomly oriented straight CNTs and also to calculate the effective material properties of functionally graded CNTRC beams. The variational iteration method was used to solve the nonlinear governing equations. [PERSON] et al. [15] claimed that using the nonlinear stress-strain relationship of graphene, a new Euler-Bernoulli beam model of SWCNT was found. In this way, the static bending and the first-order mode forced vibrations of SWCNT were proposed. The linear dynamic behavior of a hybrid nanocomposite polymer beam reinforced via carbon fibers, the Halpin-Tsai model, and exponential shear deformation beam theory were examined by [PERSON] et al. [16]. To solve the governing equations and dynamic instability, the differential quadrature method and Bolotin procedure were used. Based on the nonlocal Timoshenko beam model, [PERSON] and [PERSON] [17] analyzed the free vibration of SWCNT resting on an elastic medium. The slenderness ratios, the atomic structures, the effects of stiffness, and the boundary conditions were derived to study the natural frequencies. [PERSON] [18] explored the axial vibration of double-walled nanotubes using a semi-analytical method. The Pasternak foundation and magnetic effects influence were analyzed, keeping in mind the vibrational frequencies. The free vibration of functionally graded nanocomposite double beams reinforced by SWCNTs was studied by [PERSON] et al. [19]. The double simply supported beams coupled by an interlayer spring and resting on the elastic foundation were considered in the thermal environments. [PERSON] et al. [20] investigated the forced vibration of composite beams reinforced by SWCNTs under the influence of a moving mass. Based on a third-order shear deformation theory, the governing equations were solved by the finite element method for three different beam models (uniform, \(\Lambda\) and X-CNT). [PERSON] et al. [21] considered wave propagation analysis in functionally graded CNTRC resting on an elastic medium. The CNT fibers were graded across the beam length from a linear to parabolic pattern. The Euler-Bernoulli beam theory was considered, and the governing equations with variable coefficients were obtained. Nonlinear vibration of a CNTRC beam, based on the modified couple stress theory, was explored by [PERSON] and [PERSON] [22], accounting for the lateral harmonic load with a damping effect. [PERSON] and [PERSON] [23] applied NSGT and Euler-Bernoulli beam theory to study nonlinear free vibration and the buckling of a functionally graded porous microbeam resting on an elastic foundation.