# Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load
## 4 Free Vibrations  
The significance of the size parameter, imperfection, and lamination scheme on the free vibrations of nonlocal beams in the postbuckling regime is investigated. A small dynamic disturbance, \(v(x,t)\), is assumed to take place around a static deflected position, \(w_{s}(x)\), such that the total deflection measured from an initial configuration is defined as follows:  
\[w(x,t)=w_{s}(x)+v(x,t) \tag{29}\]  
where the static position, \(w_{s}(x)\), has been already defined for a given pre-load in the previous section. The linear free vibration problem in terms of the small dynamic disturbance, \(v(x,t)\), can be obtained by substituting Equation (25) into the governing equation, Equation (16). The result is  
\[\left(1-\overline{\mu}\frac{\partial^{2}}{\partial x^{2}}\right)\!\frac{ \partial^{2}v}{\partial t^{2}}+\left(1-\Lambda\ \overline{\mu}\right)\!\frac{\partial^{4}v}{\partial x^{4}}+\Lambda\frac{ \partial^{2}v}{\partial x^{2}}=\left(\alpha\int_{0}^{1}w_{s}^{\prime}\frac{ \partial v}{\partial x}dx\ \right)\left(1-\overline{\mu}\frac{\partial^{2}}{ \partial x^{2}}\ \right)\!w_{s}^{\prime\prime} \tag{30}\]  
We note that the terms that have definite integrals can be dealt with as constants for any given functions, \(w_{s}(x)\) and \(v(x,t)\). Assuming a harmonic solution in the form of \(v(x,t)=\phi(x)e^{i\omega t}\), and solving Equation (26), yield the following general solution  
\[\phi(x)=c_{1}\cos s_{1}x+c_{2}\sin s_{1}x+c_{3}\cosh s_{2}x+c_{4}\text{sinhs}_ {2}x+c_{5}\cos 2\ \pi x \tag{31}\]  
where \(s_{1}\) and \(s_{2}\) are given by  
\[s_{1,2}=\left[\frac{\pm\big{(}\Lambda+\omega^{2}\overline{\mu}\big{)}+\sqrt{ \Lambda^{2}+4\omega^{2}-2\Lambda\omega^{2}\overline{\mu}+\omega^{4}\overline {\mu}^{2}}}{2(1-\Lambda\overline{\mu})}\right]^{\frac{1}{2}} \tag{32}\]  
where \(\phi(x)\) is the mode shape and \(\omega\) is its frequency. This demands that the solution given by Equation (27) satisfies the boundary conditions and the equation of motion yields an eigenvalue problem that can be solved for the natural frequency, \(\omega\), and its associated mode shape. For more details, the reader is referred to [PERSON] and [PERSON] [72]. Before the numerical results are presented, it is helpful to investigate the significance of the size parameter, \(\overline{\mu}\), on the free vibrations to gain more insight. The size parameter, as can be seen from Equation (26), has an effect on the inertia, bending stiffness, and forcing terms. It can be noticed that the size parameter reduces the stiffness knowing that the constant, \(\Lambda\), is always positive. This means that the size effect has a kind of softening, which will in turn reduce the natural frequencies. The softening effect is noted from Figure 10, which represents the variation of the fundamental natural frequency with the pre-stress load for a unidirectional/cross-ply laminated beam. As the figure implies, for an initially straight multilayer nanobeam (\(a=0\)), the natural frequency decreases as the pre-stress load increases in the prebuckling domain. This trend continues until the onset of buckling, where the fundamental frequency approaches zero. After that, the natural frequency increases as the pre-stress load is increased. This observation is noted for both classical and size-dependent nanobeams. The nonlocal effect on higher modes of perfect beams is further investigated and shown in Figure 10 for the third and fifth vibration modes, respectively. A greater reduction in the higher frequencies is noticed with the consideration of the size effect. However, the rate of change in the natural frequency in both the pre and postbuckling states is much less compared to the first mode. It is worth noting that for perfect beams, the natural frequencies of the even modes do not change with the pre-stress load in the postbuckling state (Nayfeh and Emam [72]).