where

$\stackrel{\u02c6}{w}(\stackrel{\u02c6}{x},\stackrel{\u02c6}{t})$ is the transverse displacements of the beam and

*ε* is a small dimensionless parameter;

*m* denotes the mass and

$\stackrel{\u02c6}{\eta}$ is the damping coefficient;

$\stackrel{\u02c6}{F}$ is the external excitation amplitude,

$\stackrel{\u02c6}{\mathrm{\Omega}}$ is the external excitation frequencies, and

${D}^{\alpha}$ denotes the fractional derivative of order

*α*. Here, also, the dot denotes partial differentiation with respect to time

$\stackrel{\u02c6}{t}$, and prime denotes the derivative with respect to spatial

$\stackrel{\u02c6}{x}$. On the other hand, it is assumed that the tension

*T* is characterized as a small periodic perturbation

$\epsilon {T}_{1}cos\mathrm{\Omega}$ on the steady-state tension

${T}_{0}$,

*i.e.*,

$T={T}_{0}+\epsilon {T}_{1}cos\mathrm{\Omega},$

(3)

where Ω is the frequency of a beam [

12]. Introducing the dimensionless parameters as

$w=\frac{\stackrel{\u02c6}{w}}{L},\phantom{\rule{2em}{0ex}}x=\frac{\stackrel{\u02c6}{x}}{L},\phantom{\rule{2em}{0ex}}t=\frac{\stackrel{\u02c6}{t}}{L}\sqrt{\frac{{T}_{0}}{\rho A}},$

(4)

we have the new dimensionless parameters

$\begin{array}{r}a=\frac{{T}_{1}}{{T}_{0}},\phantom{\rule{2em}{0ex}}\overline{\eta}=\frac{\stackrel{\u02c6}{\eta}}{{L}^{\alpha -2}}\sqrt{\frac{{T}_{0}^{\alpha -2}}{{m}^{\alpha}}}\phantom{\rule{2em}{0ex}}(\overline{\eta}=\epsilon \eta ),\phantom{\rule{2em}{0ex}}{\mathrm{\Omega}}_{1}=\mathrm{\Omega}L\sqrt{\frac{m}{{T}_{0}}},\\ {\mathrm{\Omega}}_{2}=\stackrel{\u02c6}{\mathrm{\Omega}}L\sqrt{\frac{m}{{T}_{0}}},\phantom{\rule{2em}{0ex}}f(x)=\frac{L}{{T}_{0}}\stackrel{\u02c6}{F}(x),\end{array}$

(5)

where

*ρ* is density,

*A* is the cross-sectional area, and

*L* is the length of the beam. Thus, the equation in the non-dimensional form is presented as

$\ddot{w}-(1+\epsilon acos{\mathrm{\Omega}}_{1}t){w}^{\u2033}+\epsilon \eta {D}^{\alpha}w=f(x)cos{\mathrm{\Omega}}_{2}t,$

(6)