The steady-state response of size-dependent functionally graded nanobeams to subharmonic excitation

Abstract

This study aims to investigate the nonlinear forced vibration of functionally graded (FG) nanobeams. It is assumed that material properties are gradually graded in the direction of thickness. Nonlocal nonlinear Euler–Bernoulli beam theory is used to derive nonlocal governing equations of motion. The linear eigenmodes of FG nanobeams are used to transform a partial differential equation of motion into a system of ordinary differential equations via the Galerkin method. The multiple scale method is used to find the governing equations of the steady-state responses of FG nanobeams excited by a distributed harmonic force with constant intensity. It is also assumed that the working frequency is close to three times greater than the lowest natural frequency. Based on the equation governing the linear natural frequencies of FG nanobeams, the influence of the small scale parameter, material composition, and stiffness of the foundation on the linear relationship among natural frequencies is studied. Results show that superharmonic response or a combination of resonances may occur as well as a subharmonic response depending on the power-law index and stiffness of the foundation. Then the governing equations of a steady-state response of FG nanobeams for four possible solutions are obtained depending on the value of the small scale parameter. It is shown that the simplest response of FG nanobeams is a subharmonic response or superharmonic response. The equations governing the frequency–response curves are obtained and the effects of the power-law index and small scale parameter on them are discussed.

Appendices

Appendix 1

The equations of motion of a simply supported FG nanobeam with length L, width b, and thickness h and immovable ends can be derived using Hamilton’s principle. In this study, based on previous research [19, 21], it is assumed that the in-plane inertia and rotary inertia are negligible:

$$\begin{aligned}&\frac{\partial \hat{{N}}}{\partial x}=0, \end{aligned}$$

(33)

$$\begin{aligned}&F-\overline{{k}}W-\overline{{c}}\frac{\partial W}{\partial t}-\frac{\partial ^{2}\hat{{M}}}{\partial x^{2}}+\hat{{N}}\frac{\partial ^{2}W}{\partial x^{2}}=\left( {\int \limits _{A_0 } {\rho \left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{2}W}{\partial t^{2}}, \end{aligned}$$

(34)

where $W=W(x,t)$ is the transverse displacement of any point on the geometric midplane of the FG nanobeam element, $\rho (z)$ is the mass density, which is functionally graded in the thickness direction, $\hat{{N}}$ is the axial normal force, $\hat{{M}}$ is the bending moment, $\overline{{k}}$ is the stiffness of the foundation, $\overline{{c}}$ is the damping coefficient of the foundation, and $F=F\left( x \right) \cos \left( {\Omega t} \right)$ is the transverse loading. $A_0$ denotes the area of the FG nanobeam cross section.

According to the Euler–Bernoulli hypothesis and von Karman type geometrical nonlinearity, the strain displacement relationship is as follows [19, 21]:

$$\begin{aligned} \varepsilon _x =\frac{\partial u_1 }{\partial x}+\frac{1}{2}\left( {\frac{\partial W}{\partial x}} \right) ^{2}, \end{aligned}$$

(35)

where $u_{1}$ is the total displacement along the x-direction given by Eq. (36):

$$\begin{aligned} u_1 \left( {x,z,t} \right) =u_0 \left( {x,t} \right) -\left( {z-z_0 } \right) \frac{\partial W}{\partial x}, \end{aligned}$$

(36)

where $u_0 \left( {x,t} \right)$ is an axial displacement of any point on the geometric midplane of the FG nanobeam element, and $z_{0}$ is the distance between the neutral surface and the geometric midplane of the FG nanobeam (Fig. 8) [14].

Fig. 8

Cross section of functionally graded beam showing distance of neutral surface from geometric midplane

According to the physical concept of a neutral surface, $z_{0}$ can be written as follows [14] (details can be found in Ref. [14]):

$$\begin{aligned} z_0 ={\int \limits _{A_0 } {zE\left( z \right) \mathrm{d}A_0 } }\Big /{\int \limits _{A_0 } {E\left( z \right) \mathrm{d}A_0 } }. \end{aligned}$$

(37)

Based on Eringen’s nonlocal elasticity, the stress–strain relationship is

$$\begin{aligned} \sigma _x -\left( {e_0 a} \right) ^{2}\nabla ^{2}\sigma _x =E\varepsilon _x , \end{aligned}$$

(38)

where $\hbox {e}_{0}\hbox {a}$ is a material length scale parameter that contains a material constant and internal characteristic length. On the other hand, the resultant axial force and resultant bending moment are ($\hat{{N}}$ and $\hat{{M}}$)

$$\begin{aligned} \hat{{N}}=\int \limits _{A_0 } {\sigma _x \mathrm{d}A_0 } , \hat{{M}}=-\int \limits _{A_0 } {\sigma _x } \left( {z-z_0 } \right) \mathrm{d}A_0. \end{aligned}$$

(39)

The stress resultants on a beam element can be obtained by substituting Eqs. (35) and (38) into Eq. (39):

$$\begin{aligned}&\hat{{N}}-\left( {e_0 a} \right) ^{2}\nabla ^{2}\hat{{N}}=\left( {\int _{A_0 } {E\left( z \right) \mathrm{d}A_0 } } \right) \left[ {\frac{\partial u}{\partial x}+\frac{1}{2}\left( {\frac{\partial W}{\partial x}} \right) ^{2}} \right] , \end{aligned}$$

(40a)

$$\begin{aligned}&\hat{{M}}-\left( {e_0 a} \right) ^{2}\nabla ^{2}\hat{{M}}=\left( {\int _{A_0 } {\left( {z-z_0 } \right) ^{2}E\left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{2}W}{\partial x^{2}}, \end{aligned}$$

(40b)

where E(z) and $\rho (z)$ defined by Eq. (41) are the Young’s modulus and specific mass density of the FG beam material, respectively.

$$\begin{aligned}&E\left( z \right) =E_1 +\left( {E_2 -E_1 } \right) \left( {\frac{2z+h}{2h}} \right) ^{\overline{{n}}}, \end{aligned}$$

(41a)

$$\begin{aligned}&\rho \left( z \right) =\rho _1 +\left( {\rho _2 -\rho _1 } \right) \left( {\frac{2z+h}{2h}} \right) ^{\overline{{n}}} , \end{aligned}$$

(41b)

where $\hbox {E}_{i}$ and $\rho _{i}$ (i $=$ 1, 2) are the Young’s modulus and the specific mass density of the two materials used in the construction of the FG beam, respectively.

The partial differential equation of the transverse motion of FG nanobeams can be derived by combining Eq. (40b) and Eq. (34) and making some simplifications:

$$\begin{aligned} \left( {e_0 a} \right) ^{2}H+\hat{{N}}\frac{\partial ^{2}W}{\partial x^{2}}-\overline{{k}}W-\overline{{c}}\frac{\partial W}{\partial t}+F\left( {x,t} \right) =\left( {\int \limits _{A_0 } {\left( {z-z_0 } \right) ^{2}E\left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{4}W}{\partial x^{4}}+\left( {\int \limits _{A_0 } {\rho \left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{2}W}{\partial t^{2}},\nonumber \\ \end{aligned}$$

(42)

where H is defined by Eq. (43):

$$\begin{aligned} H=\overline{{c}}\frac{\partial ^{3}W}{\partial x^{2}\partial t}+\overline{{k}}\frac{\partial ^{2}W}{\partial x^{2}}-\frac{\partial ^{2}F}{\partial x^{2}}-\hat{{N}}\frac{\partial ^{4}W}{\partial x^{4}}+\left( {\int \limits _{A_0 } {\rho \left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{4}W}{\partial x^{2}\partial t^{2}}. \end{aligned}$$

(43)

On the basis of Eq. (33), one can conclude that $\nabla ^{2}\hat{{N}}$ is zero. Therefore, Eq. (40a) is simplified to the relationship between the axial force $\hat{{N}}$ and displacement components of the midplane of the FG beam (W and $u_{0}$) as follows [19, 21]:

$$\begin{aligned} \hat{{N}}=\left( {\int \limits _{A_0 } {E\left( z \right) \mathrm{d}A_0 } } \right) \left[ {\frac{\partial u_0 }{\partial x}+\frac{1}{2}\left( {\frac{\partial W}{\partial x}} \right) ^{2}} \right] . \end{aligned}$$

(44a)

Integrating Eq. (44a) yields [19, 21]

$$\begin{aligned} \int \limits _0^L {\hat{{N}}\mathrm{d}x} =\int \limits _0^L {\left( {\left( {\int \limits _{A_0 } {E\left( z \right) \mathrm{d}A_0 } } \right) \left[ {\frac{\partial u_0 }{\partial x}+\frac{1}{2}\left( {\frac{\partial W}{\partial x}} \right) ^{2}} \right] } \right) } \mathrm{d}x \end{aligned}$$

(45a)

or

$$\begin{aligned} \hat{{N}}L=\left( {\int \limits _{A_0 } {E\left( z \right) \mathrm{d}A_0 } } \right) \left[ {u_0 \left( L \right) -u_0 \left( 0 \right) +\int \limits _0^L {\frac{1}{2}\left( {\frac{\partial W}{\partial x}} \right) ^{2}\mathrm{d}x} } \right] . \end{aligned}$$

(45b)

The boundary values of the axial displacement of nanobeams are [19, 21]

$$\begin{aligned} u_0 \left( 0 \right) =0 ,\quad u_0 \left( L \right) =0. \end{aligned}$$

(46)

The relationship between the axial force $\hat{{N}}$ and the transverse displacement of the midplane of nanobeams can be obtained by substituting for boundary conditions from Eq. (46) into Eq. (45b):

$$\begin{aligned} \hat{{N}}=+\frac{1}{2L}\left( {\int \limits _{A_0 } {E\left( z \right) } \mathrm{d}A_0 } \right) \int \limits _0^L {\left( {\frac{\partial W}{\partial x}} \right) ^{2}\mathrm{d}x}. \end{aligned}$$

(47)

Substituting the $\hat{{N}}$ from Eq. (47) into Eqs. (42) and (43), one can obtain the governing equation of the nonlinear forced lateral vibration of FG nanobeams as follows:

$$\begin{aligned} \left( {e_0 a} \right) ^{2}H+\hat{{N}}\frac{\partial ^{2}W}{\partial x^{2}}-\overline{{k}}W-\overline{{c}}\frac{\partial W}{\partial t}+F\left( {x,t} \right) =\left( {\int \limits _{A_0 } {\left( {z-z_0 } \right) ^{2}E\left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{4}W}{\partial x^{4}}+\left( {\int \limits _{A_0 } {\rho \left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{2}W}{\partial t^{2}},\nonumber \\ \end{aligned}$$

(48)

where H is defined by Eq. (49):

$$\begin{aligned} H=\overline{{c}}\frac{\partial ^{3}W}{\partial x^{2}\partial t}+\overline{{k}}\frac{\partial ^{2}W}{\partial x^{2}}-\frac{\partial ^{2}F}{\partial x^{2}}-\hat{{N}}\frac{\partial ^{4}W}{\partial x^{4}}+\left( {\int \limits _{A_0 } {\rho \left( z \right) \mathrm{d}A_0 } } \right) \frac{\partial ^{4}W}{\partial x^{2}\partial t^{2}}, \end{aligned}$$

(49)

and $\hat{{N}}$ is

$$\begin{aligned} \hat{{N}}=+\frac{1}{2L}\left( {\int \limits _{A_0 } {E\left( z \right) } \mathrm{d}A_0 } \right) \int \limits _0^L {\left( {\frac{\partial W}{\partial x}} \right) ^{2}\mathrm{d}x} . \end{aligned}$$

(50)

The following dimensionless variables are used to simplify the parametric studies:

$$\begin{aligned} \overline{{x}}= & {} \frac{x}{L},\quad \overline{{W}}=\frac{W}{r},\quad \overline{{t}}=t\sqrt{D/{\rho _e L^{4}}}, \nonumber \\ D= & {} b\int \limits _{-\frac{h}{2}}^{\frac{h}{2}} {\left( {z-z_0 } \right) ^{2}E\left( z \right) \mathrm{d}z, \rho _e =b\int \limits _{-\frac{h}{2}}^{\frac{h}{2}} {\rho \left( z \right) \mathrm{d}z} } , A=b\int \limits _{-\frac{h}{2}}^{\frac{h}{2}} {E\left( z \right) \mathrm{d}z} , r=\sqrt{{bh^{3}}/{12(bh)}}. \end{aligned}$$

(51)

Then, the governing partial deferential equation of motion changes to

$$\begin{aligned}&-\frac{L^{4}\left( {e_0 a} \right) ^{2}}{rD}\overline{{H}}+\frac{\partial ^{2}\overline{{W}}}{\partial \overline{{t}}^{2}}+\frac{\partial ^{4}\overline{{W}}}{\partial \overline{{x}}^{4}}+\frac{\overline{{k}}L^{4}}{D}\overline{{W}}+\frac{\overline{{c}}L^{2}}{\sqrt{D\rho _e }}\frac{\partial \overline{{W}}}{\partial \overline{{t}}} -\left( {\frac{Ar^{2}}{2D}\int \limits _0^1 {\left( {\frac{\partial \overline{{W}}}{\partial \overline{{x}}}} \right) ^{2}\mathrm{d}\overline{{x}}} } \right) \frac{\partial ^{2}\overline{{W}}}{\partial \overline{{x}}^{2}}=F\frac{L^{4}}{rD}, \end{aligned}$$

(52)

where

$$\begin{aligned} \overline{{H}}= & {} -\frac{1}{L^{2}}\frac{\partial ^{2}F}{\partial \overline{{x}}^{2}}+\frac{\overline{{k}}r}{L^{2}}\frac{\partial ^{2}\overline{{W}}}{\partial \overline{{x}}^{2}}+\frac{\overline{{c}}r}{L^{2}}\sqrt{\frac{D}{\rho _e L^{4}}}-\left[ {\frac{Ar^{3}}{2L^{6}}\int \limits _0^1 {\left( {\frac{\partial \overline{{W}}}{\partial \overline{{x}}}} \right) ^{2}\mathrm{d}\overline{{x}}} } \right] \frac{\partial ^{4}\overline{{W}}}{\partial \overline{{x}}^{4}} + \frac{Dr}{L^{6}}\frac{\partial ^{4}\overline{{W}}}{\partial \overline{{x}}^{2}\partial \overline{{t}}^{2}}. \end{aligned}$$

(53)

Appendix 2

According to Eq. (10), it can be found that the secular terms are eliminated from $q_{11}$, $q_{k1}$, and $q_{n1} (n\ne 1,k)$ if

$$\begin{aligned}&2i\overline{{\omega }}_1 \left( {{A}'_1 +\hat{{C}}_1 A_1 } \right) +2a_1 A_1 \sum _{m=1}^N {m^{2}\left( {A_m \overline{{A}}_m +\Lambda _m^2 } \right) } +3a_1 \overline{{A}}_1^2 \Lambda _1 \exp \left( {i\sigma T_2 } \right) +4a_1 \Lambda _1^2 A_1 +a_1 \overline{{A}}_1 A_1^2 =0, \end{aligned}$$

(54)

$$\begin{aligned}&2i\overline{{\omega }}_k \left( {{A}'_k +\hat{{C}}_k A_k } \right) +2a_k A_k \sum _{m=1}^N {m^{2}\left( {A_m \overline{{A}}_m +\Lambda _m^2 } \right) } +3a_1 \overline{{A}}_1^2 \Lambda _1 \exp \left( {i\sigma T_2 } \right) +4a_k k^{2}\Lambda _k^2 A_k +k^{2}a_k \overline{{A}}_k A_k^2 \nonumber \\&\qquad +\,3a_k \Lambda _k \exp \left( {i\sigma _1 T_2 } \right) \sum _{m=1}^N {m^{2}\Lambda _m^2 } =0, \end{aligned}$$

(55)

$$\begin{aligned}&2i\overline{{\omega }}_n \left( {{A}'_n +\hat{{C}}_n A_n } \right) +2a_n A_n \sum _{m=1}^N {m^{2}\left( {A_m \overline{{A}}_m +\Lambda _m^2 } \right) } +4a_n n^{2}\Lambda _n^2 A_n +n^{2}a_n \overline{{A}}_n A_n^2 =0. \end{aligned}$$

(56)

Let

$$\begin{aligned} A_m =\frac{1}{2}\alpha _m \exp (i\beta _m ), \end{aligned}$$

(57)

where $\alpha _m \left( {T_2 } \right)$ and $\beta _m \left( {T_2 } \right)$ are both real. Substituting Eq. (57) into Eq. (54), Eq. (55), and Eq. (56) and separating the result into real and imaginary parts, one obtains

$$\begin{aligned}&{\alpha }'_1 +\hat{{C}}_1 \alpha _1 +\frac{3}{4}\frac{a_1 \Lambda _1 \alpha _1^2 }{\overline{{\omega }}_1 }\sin \gamma _1 =0, \end{aligned}$$

(58)

$$\begin{aligned}&-\alpha _1 {\beta }'_1 +a_1 \alpha _1 \sum _{m=2}^N {m^{2}\left( {\frac{\alpha _m^2 }{4}+\Lambda _m^2 } \right) } +\frac{3}{4}a_1 \alpha _1^2 \Lambda _1 \cos \left( {\sigma T_2 -3\beta _1 } \right) +3a_1 \Lambda _1^2 \alpha _1 +\frac{3}{8}a_1 \alpha _1^3 =0, \end{aligned}$$

(59)

$$\begin{aligned}&{\alpha }'_k \overline{{\omega }}_k +\hat{{C}}_k \alpha _k \overline{{\omega }}_k +3a_k \Lambda _k \sum _{m=1}^N {m^{2}\Lambda _m^2 \sin \gamma _2 } =0, \end{aligned}$$

(60)

$$\begin{aligned}&-\alpha _k {\beta }'_k \overline{{\omega }}_k +3a_k \Lambda _k \sum _{m=1}^N {m^{2}\Lambda _m^2 \cos \gamma _2 +a_k \alpha _k \sum _{m=1}^N {m^{2}\left( {\frac{\alpha _m^2 }{4}+\Lambda _m } \right) } +2k^{2}a_k \Lambda _k^2 \alpha _k +\frac{k^{2}}{8}a_k \alpha _k^3 } =0, \end{aligned}$$

(61)

$$\begin{aligned}&\overline{{\omega }}_n \left( {{\alpha }'_n +\hat{{C}}_n \alpha _n } \right) =0, \end{aligned}$$

(62)

$$\begin{aligned}&-\overline{{\omega }}_n \alpha _n {\beta }'_n +a_n \alpha _n \sum _{m=1}^N {\frac{m^{2}}{4}\left( {\alpha _m^2 +4\Lambda _m^2 } \right) } +2a_n n^{2}\Lambda _n^2 \alpha _n +\frac{1}{8}n^{2}a_n \alpha _n^3 =0, \end{aligned}$$

(63)

where $\sigma T_2 -3\beta _1 =\gamma _1$, $\sigma _1 T_2 -\beta _k =\gamma _2$, and $\left( \right) ^{\prime }={d}/{dT_2 }$.

The solution of Eq. (62) shows that

$$\begin{aligned} \alpha _n \propto \exp (-\hat{{C}}_n T_2 ). \end{aligned}$$

(64)

Equation (64) clearly reveals that all $\alpha _n (n\ne 1,k)$ decay. Hence, the steady-state motion (${\alpha }'_1 =0,{\alpha }'_k =0,{\gamma }'_1 =0,{\gamma }'_k =0)$ can be expressed as

$$\begin{aligned}&\hat{{C}}_1 \alpha _1 \overline{{\omega }}_1 +\frac{3}{4}a_1 \Lambda _1 \alpha _1^2 \sin \gamma _1 =0, \end{aligned}$$

(65)

$$\begin{aligned}&-\frac{1}{3}\sigma \alpha _1 \overline{{\omega }}_1 +a_1 \alpha _1 \sum _{m=2}^N {m^{2}\Lambda _m^2 } +\frac{3}{4}a_1 \alpha _1^2 \Lambda _1 \cos \left( {\gamma _1 } \right) +\frac{k^{2}}{4}\alpha _1 a_1 \alpha _k^2 +3a_1 \Lambda _1^2 \alpha _1 +\frac{3}{8}a_1 \alpha _1^3 =0, \end{aligned}$$

(66)

$$\begin{aligned}&\hat{{C}}_k \alpha _k \overline{{\omega }}_k +a_k \Lambda _k \sum _{m=1}^N {m^{2}\Lambda _m^2 \sin \gamma _2 } =0, \end{aligned}$$

(67)

$$\begin{aligned}&-\alpha _k \sigma _1 \overline{{\omega }}_k +a_k \Lambda _k \sum _{m=1}^N {m^{2}\Lambda _m^2 \cos \gamma _2 +\frac{3}{8}a_k k^{2}\alpha _k^3 +\frac{1}{4}a_k \alpha _k \alpha _1^2 +a_k \alpha _k \sum _{m=1}^N {m^{2}\Lambda _m^2 } +2k^{2}a_k \Lambda _k^2 \alpha _k } =0. \end{aligned}$$

(68)

Combining Eq. (65) with Eq. (66) leads to

$$\begin{aligned} \left( \frac{3}{4}a_1 {\Lambda }_1 {\alpha }_1^2 \right) ^{2}=\left( {-\frac{1}{3}\sigma \alpha _1 \overline{{\omega }}_1 +a_1 \alpha _1 \sum _{m=2}^N {m^{2}{\Lambda }_m^2 } +\frac{k^{2}}{4}\alpha _1 a_1 \alpha _k^2 +3a_1 \Lambda _1^2 \alpha _1 +\frac{3}{8}a_1 \alpha _1^3 } \right) ^{2}+\left( {\hat{{C}}_1 \alpha _1 \overline{{\omega }}_1 } \right) ^{2}. \end{aligned}$$

(69)

Eliminating $\gamma _2$ from Eq. (67) and Eq. (68) yields

$$\begin{aligned}&\left( {a_k \Lambda _k \sum _{m=1}^N {m^{2}\Lambda _m^2 } } \right) ^{2}\!=\!\left( {-\alpha _k \sigma _1 \overline{{\omega }}_k +\frac{3}{8}a_k k^{2}\alpha _k^3 +\frac{1}{4}a_k \alpha _k \alpha _1^2 +a_k \alpha _k \sum _{m=1}^N {m^{2}\Lambda _m^2 } +2k^{2}a_k \Lambda _k^2 \alpha _k } \right) ^{2}+\left( {\hat{{C}}_k \alpha _k \overline{{\omega }}_k } \right) ^{2}.\nonumber \\ \end{aligned}$$

(70)

According to Eqs. (69) and (70), these two solutions are possible: either $\alpha _1 =0$ and $\alpha _k \ne 0$ or $\alpha _1 \ne 0$ and $\alpha _k \ne 0$.

The following equation expresses the time-dependent lateral deflection of a vibrating FG nanobeam:

$$\begin{aligned} \overline{{W}}\left( {\overline{{x}},\overline{{t}}} \right)= & {} \varepsilon \alpha _1 \cos \left( {\frac{1}{3}\overline{{\Omega }}\overline{{t}}-\frac{1}{3}\gamma _1 } \right) \sin \left( {\pi \overline{{x}}} \right) +\varepsilon \alpha _k \cos \left( {3\overline{{\Omega }}\overline{{t}}-\gamma _2 } \right) \sin \left( {k\pi \overline{{x}}} \right) \nonumber \\&+\,2\varepsilon \sum _{n=1}^N {\Lambda _n \cos \left( \, {\overline{{\Omega }}\overline{{t}}} \right) } \sin \left( {n\pi \overline{{x}}} \right) +O\big ( {\varepsilon ^{3}} \big ). \end{aligned}$$

(71)

Appendix 3

It can be found that Eqs. (58)–(63) govern the conditions in which secular terms are eliminated from $q_{11},q_{k1}$ and $q_{n1}(n\ne 1,k,p,q)$.

The following equations must be satisfied to eliminate the secular terms from $q_{p1}$ and $q_{q1}$:

$$\begin{aligned}&2i\overline{{\omega }}_p \left( {{A}'_p +\hat{{C}}_p A_p } \right) +2a_p A_p \sum _{m=1}^N {m^{2}\left( {A_m \overline{{A}}_m +\Lambda _m^2 } \right) } +4a_p p^{2}\Lambda _p^2 A_p +p^{2}a_p \overline{{A}}_p A_p^2 \nonumber \\&\quad +\,2a_p \Lambda _q \Lambda _p \overline{{A}}_q \exp \left( {\sigma _2 T_2 } \right) =0, \end{aligned}$$

(72)

$$\begin{aligned}&2i\overline{{\omega }}_q \left( {{A}'_q +\hat{{C}}_q A_q } \right) +2a_q A_q \sum _{m=1}^N {m^{2}\left( {A_m \overline{{A}}_m +\Lambda _m^2 } \right) } +4a_q q^{2}\Lambda _q^2 A_q +q^{2}a_q \overline{{A}}_q A_q^2 \nonumber \\&\quad +\,2a_q \Lambda _q \Lambda _p \overline{{A}}_p \exp \left( {\sigma _2 T_2 } \right) =0. \end{aligned}$$

(73)

The following equations can be found by introducing the polar notation represented by Eq. (57):

$$\begin{aligned}&\overline{{\omega }}_p \left( {{\alpha }'_p +\hat{{C}}_p \alpha _p } \right) +a_p \Lambda _p \Lambda _q \alpha _q \sin \left( {\gamma _3 } \right) =0, \end{aligned}$$

(74)

$$\begin{aligned}&\quad -\,\overline{{\omega }}_p \alpha _p {\beta }'_p +a_p \alpha _p \sum _{m=1}^N {\frac{m^{2}}{4}\left( {\alpha _m^2 +4\Lambda _m^2 } \right) } +2a_p p^{2}\Lambda _p^2 \alpha _p +\frac{1}{8}p^{2}a_p \alpha _p^3 +a_p \Lambda _p \Lambda { }_q\alpha _q \cos \left( {\gamma _3 } \right) =0, \end{aligned}$$

(75)

$$\begin{aligned}&\overline{{\omega }}_q \left( {{\alpha }'_q +\hat{{C}}_q \alpha _q } \right) +a_q \Lambda _q \Lambda _p \alpha _p \sin \left( {\gamma _3 } \right) =0, \end{aligned}$$

(76)

$$\begin{aligned}&\quad -\,\overline{{\omega }}_q \alpha _q {\beta }'_q +a_q \alpha _q \sum _{m=1}^N {\frac{m^{2}}{4}\left( {\alpha _m^2 +4\Lambda _m^2 } \right) } +2a_q q^{2}\Lambda _q^2 \alpha _q +\frac{1}{8}q^{2}a_q \alpha _q^3 +a_q \Lambda _q \Lambda { }_p\alpha _p \cos \left( {\gamma _3 } \right) =0, \end{aligned}$$

(77)

in which $\gamma _3 =\sigma _2 T_2 -\beta _p -\beta _q$.

As mentioned earlier, except for $\alpha _1 , \alpha _k , \alpha _p$ and $\alpha _q$, the remaining $\alpha _n \left( {n\ne 1,k,p,q} \right)$ decay with time. Thus, the following equations govern the steady-state response of FG nanobeams:

$$\begin{aligned}&\overline{{\omega }}_p \hat{{C}}_p \alpha _p +a_p \Lambda _p \Lambda _q \alpha _q \sin \left( {\gamma _3 } \right) =0, \end{aligned}$$

(78)

$$\begin{aligned}&\overline{{\omega }}_q \hat{{C}}_q \alpha _q +a_q \Lambda _q \Lambda _p \alpha _p \sin \left( {\gamma _3 } \right) =0, \end{aligned}$$

(79)

$$\begin{aligned}&\sigma _2 ={\beta }'_p +{\beta }'_q , \end{aligned}$$

(80)

$$\begin{aligned} {\beta }'_p= & {} \frac{1}{\overline{{\omega }}_p \alpha _p }\left( {a_p \alpha _p \sum _{m=1}^N {m^{2}\Lambda _m^2 \frac{m^{2}}{4}+} \frac{q^{2}}{4}a_p \alpha _p \alpha _q^2 +\frac{1}{4}a_p \alpha _p \alpha _1^2 +\frac{k^{2}}{4}a_p \alpha _p \alpha _k^2 +2a_p p^{2}\Lambda _p^2 \alpha _p } \right) \nonumber \\&+\,\frac{1}{\overline{{\omega }}_p \alpha _p }\left( {\frac{3}{8}p^{2}a_p \alpha _p^3 +a_p \Lambda _p \Lambda { }_q\alpha _q \cos \left( {\gamma _3 } \right) } \right) , \end{aligned}$$

(81)

$$\begin{aligned} {\beta }'_q= & {} \frac{1}{\overline{{\omega }}_q \alpha _q }\left( {a_q \alpha _q \sum _{m=1}^N {m^{2}\Lambda _m^2 \frac{m^{2}}{4}+} \frac{p^{2}}{4}a_q \alpha _q \alpha _p^2 +\frac{1}{4}a_q \alpha _q \alpha _1^2 +\frac{k^{2}}{4}a_q \alpha _q \alpha _k^2 +2q^{2}a_q \Lambda _q^2 \alpha _q } \right) \nonumber \\&+\,\frac{1}{\overline{{\omega }}_q \alpha _q }\left( {\frac{3}{8}q^{2}a_q \alpha _q^3 +a_q \Lambda _q \Lambda { }_p\alpha _p \cos \left( {\gamma _3 } \right) } \right) , \end{aligned}$$

(82)

$$\begin{aligned}&\hat{{C}}_1 \alpha _1 \overline{{\omega }}_1 +\frac{3}{4}a_1 \Lambda _1 \alpha _1^2 \sin \gamma _1 =0, \end{aligned}$$

(83)

$$\begin{aligned}&\quad -\,\frac{1}{3}\sigma \alpha _1 \overline{{\omega }}_1 +a_1 \alpha _1 \sum _{m=2}^N {m^{2}\Lambda _m^2 } +\frac{3}{4}a_1 \alpha _1^2 \Lambda _1 \cos \left( {\gamma _1 } \right) +\frac{k^{2}}{4}\alpha _1 a_1 \alpha _k^2 +3a_1 \Lambda _1^2 \alpha _1 +\frac{3}{8}a_1 \alpha _1^3 \nonumber \\&\quad +\,\frac{3}{4}p^{2}\alpha _1 a_1 \alpha _p^2 +\frac{3}{4}q^{2}\alpha _1 a_1 \alpha _q^2 =0, \end{aligned}$$

(84)

$$\begin{aligned}&\hat{{C}}_k \alpha _k \overline{{\omega }}_k +a_k \Lambda _k \sum _{m=1}^N {m^{2}\Lambda _m^2 \sin \gamma _2 } =0, \end{aligned}$$

(85)

$$\begin{aligned}&\quad -\,\alpha _k \sigma _1 \overline{{\omega }}_k +a_k \Lambda _k \sum _{m=1}^N {m^{2}\Lambda _m^2 \cos \gamma _2 +\frac{3}{8}a_k k^{2}\alpha _k^3 +\frac{1}{4}a_k \alpha _k \alpha _1^2 +a_k \alpha _k \sum _{m=1}^N {m^{2}\Lambda _m^2 } +2k^{2}a_k \Lambda _k^2 \alpha _k } \nonumber \\&\quad +\,\frac{p^{2}}{4}\alpha _k a_k \alpha _p^2 +\frac{q^{2}}{4}\alpha _k a_k \alpha _q^2 =0. \end{aligned}$$

(86)

According to Eqs. (78), (79), and (83), the possible solutions are

• (a) $\alpha _1 =\alpha _p =\alpha _q =0$, and $\alpha _k \ne 0$.

• (b) $\alpha _1 \ne 0$, and $\alpha _k \ne 0$, but $\alpha _p =\alpha _q =0$.

• (c) $\alpha _1 =0$, and $\alpha _k \ne 0$, $\alpha _p \ne 0$, and $\alpha _q \ne 0$.

• (d) $\alpha _1 \ne 0$, $\alpha _k \ne 0$,$\alpha _p \ne 0$, and $\alpha _q \ne 0$.

The lateral deflection of vibrating FG nanobeams can be represented by the following equation:

$$\begin{aligned} \overline{{W}}\left( {\overline{{x}},\overline{{t}}} \right)&=\varepsilon \alpha _1 \cos \left( {\frac{1}{3}\overline{{\Omega }}\overline{{t}}-\frac{1}{3}\gamma _1 } \right) \sin \left( {\pi \overline{{x}}} \right) +\varepsilon \alpha _k \cos \left( {3\overline{{\Omega }}\overline{{t}}-\gamma _2 } \right) \sin \left( {k\pi \overline{{x}}} \right) +\varepsilon \alpha _p \cos \left( {\overline{{\omega }}_p \overline{{t}}+\beta _P } \right) \sin \left( {p\pi \overline{{x}}} \right) \nonumber \\&\quad +\,\varepsilon \alpha _q \cos \left( {\overline{{\omega }}_q \overline{{t}}+\beta _q } \right) \sin \left( {q\pi \overline{{x}}} \right) +2\varepsilon \sum _{n=1}^N {\Lambda _n \cos \left( {\overline{{\Omega }}\overline{{t}}} \right) } \sin \left( {n\pi \overline{{x}}} \right) +O\big ( {\varepsilon ^{3}} \big ). \end{aligned}$$

(87)