Document Type : Research Paper
Authors
School of Engineering, Damghan University, Damghan, Iran
Abstract
Keywords
Main Subjects
Nonlinear Torsional Vibration of SizeDependent Functionally Graded Rods
School of Engineering, Damghan University, Damghan, Iran
KEYWORDS 

ABSTRACT 
Torsional vibration; VonKármán type nonlinearity; Surface elasticity theory; Internal resonances; Multiple scale method. 
This study aims to investigate the effect of functionally graded materials (FGMs) on the internal resonances of nanorods in torsional vibration. The vonKármán type nonlinearity is considered and the governing equation of motion is derived using Hamilton's principle based on the surface elasticity theory. It is assumed that the properties of the functionally graded (FG) nanorod vary through the radius direction based on powerlaw distribution. Then, the multimode Galerkin method is implemented to convert the partial differential equation to an ordinary differential one. In the next step, the method of multiple scales is used to derive the natural frequencies as well as the conditions in which the internal resonances occur. The results are presented for two types of end conditions, fixedfixed and fixedfree, and the effects of variations of various parameters like length, radius, and amplitude of vibration on natural frequencies are investigated. This research shows that functionally graded materials differ in the state of happening the internal resonances in the presence of the surface energy. 
Functionally graded materials (FGMs) are nonhomogeneous materials made up by changing volume fractions of two (or more) various materials in the chosen spatial directions. The gradual change in materials results in an inhomogeneous composite with smooth and constant mechanical, electrical, and thermal properties. These variations remove interplane issues and lead to stress distribution with a smooth profile, upper fracture toughness, and better thermal resistance, which captured significant attention in numerous engineering applications. Due to these excellent properties, FG structures have been used in different engineering fields like mechanical, aerospace, chemical, and biomechanics. [113].
By growing in material fabrication and nanostructures, FG materials are increasingly used in many micro and nanoapplications such as micro/nanoelectromechanical systems (M/NEMS), micro switches, and microelectronics [1422]. In this matter, numerous experimental investigations have been done to study the mechanical behavior of micro and nanostructures [2325]. The results from the experimental studies have shown that since classical continuum mechanics theories are not sizedependent, various novel continuum mechanics theories are needed to study nanoscaled structures and predict their behavior. Thus, to account for the intrinsic characteristics of materials at micro and nanostructures various theories have been suggested to explain the elastic behaviors of these micro and nanoscale systems such as surface elasticity theory, nonlocal elasticity theory, modified couple stress theory, and strain gradient theory. As surface energy is less significant than bulk energy at the macroscale, its effect is not taken into account. The surface effects become significant because nanoscaled objects have a high surfacetovolume ratio. Consequently, it is important to consider surface energy while doing a mechanical study of nanostructures. Gurtin et al. [26, 27] presented a mathematical theory for the accurate prediction of mechanical behaviors of nanoscaled structures to examine the impact of surface stress on the mechanics of nanostructures. As a result, in recent years, the number of research reports on this subject dramatically grew [2833].
Baron et al. developed a continuous model for nanobeams that takes into account both surface effects and material heterogeneity [34]. A theoretical model was presented by Wang and Feng to study the effects of surface elasticity and residual surface tension on the natural frequency of microbeams [35]. The Gurtin and Murdoch theory was used by Ansari and Sahmani to study the buckling and bending of nanobeams. From their work, explicit formulas for various beam theories were derived [36]. A modified continuum model was developed by Ansari et al. to predict the postbuckling deflection of nanobeams [37]. The generalized differential quadrature (GDQ) method was used to solve governing differential equations. Abbasion et al. [38] presented a comprehensive model to study how surface elasticity and residual surface tension affect the natural frequency of microbeam flexural vibrations when shear deformation and rotary inertia effects are considered. Based on the nonlocal elasticity theory, Wang presented an analytical model to predict surface effects on fluidconveying nanotubes' free vibration [39]. Their findings showed that the surface effects with positive elastic constant or positive residual surface tension tend to increase critical flow velocity and the natural frequency. The influence of surface effects on the vibration of nanotubes was studied by Farshi et al. based on the Timoshenko beam model [40]. The influence of surface elasticity and surface stress on the static bending of nanowires was investigated by He and Lilley [41]. Three different boundary conditions i.e. clampedfree, simply supported, and clamped–clamped was taken into account using the Euler–Bernoulli beam theory. Civalek et al. studied the sizedependent stability analysis of restrained nanobeam with functionally graded material via nonlocal Euler–Bernoulli beam theory using the Fourier series [42]. Uzun and Yayli reformulated a new stability model for the nanosized beam resting on a oneparameter elastic foundation. The stability solution was based on the nonlocal strain gradient elasticity theory [43].
In recent times, the influence of surface effect on the nonlinear free vibration behaviors of nanobeams has been done in some research. The nonlinear free vibration of nanobeams was investigated by Nazemnezhad et al. [44]. They considered surface effects and used Euler–Bernoulli beam theory. Based on the GurtineMurdoch continuum theory, the nonlinear free vibration behavior of Timoshenko nanobeams was studied by Ansari et al. [45]. Nonlinear free vibration of functionally graded nanobeams was studied by Asgharifard Sharabiani et al. using the EulerBernoulli beam theory [46]. Yayli investigated various points including buckling, thermal buckling, axial vibration, lateral vibration, and longitudinal vibration [4753].
Torsional vibration becomes significant in some devices, such as nanoelectromechanical systems, nanoscaled shafts, and nano servomotors as nanotubes are exposed to external torques. Studies on the free torsional vibration of nanotubes are few. Lim et al. [54] developed a new elastic nonlocal stress model as well as analytical solutions for the torsional dynamics of circular nanorods/nanotubes. Free torsional vibration behaviors of nanotubes made of a bidirectional functionally graded (FG) material with properties that changed continuously along the radius and length directions investigated by Li and Hi [55]. Based on the nonlocal elasticity theory, Murmu et al. [56] examined the torsional vibration of singlewalled carbon nanotube–buckyball systems. One end of the singlewalled carbon nanotube (SWCNT) was fixed and the other end was used to attach the Buckyball. Civalek et al. investigated static and free torsional vibration of functionally graded (FG) nanorods using the Fourier sine series and boundary conditions were described by the two elastic torsional springs at the ends [57]. Sizedependent static and free torsional vibration responses of functionally graded porous nanotubes were examined by Uzun and Yayli by using the Fourier sine series and Stokes’ transformation [43, 58]. Nazemnezhad and Fahimi studied the torsional vibration of nanobeams with a periphery crack and various end conditions [59]. Various boundary conditions, the surface shear modulus, the surface stress, and the surface density are considered on the torsional vibration of nanobeams. In another research, Nazemnezhad et al. [60] studied the effects of surface energy on the nonlinear torsional vibrations and internal resonances of nanorods. In the research, the secondorder term for the angle of rotation is considered for the displacement field. They reported the conditions wherein the internal resonances occur. The effect of the surrounding elastic matrix on the axial and torsional vibrations of embedded singlewalled boron nitride nanotube (SWBNNT) was studied by Uzun et al. The SWBNNT was modeled as a nanorod and the nonlocal strain gradient theory was utilized to derive the sizedependent equation of motion. Also, a oneparameter foundation model was employed to simulate the surrounding elastic matrix [61, 62]. The torsional vibration of nanorods with torsional elastic boundary conditions via nonlocal elasticity theory was presented by Yayli [63].
The above literature survey shows that the linear and nonlinear torsional vibrations of nanorods in the presence of surface energy are investigated. The material properties of nanorods were assumed to be homogenous. Therefore there is a crucial question: What is the torsional behavior of nanorods in the case of functionally graded materials? It can be said that answering this question is the main goal of the present study.
Figure 1 shows an FG nanorod with inner radius R_{i}_{ }and outer radius R_{o} made from a graded mixture of aluminum and silicon.
The inner surface (r = R_{i}) of the FG section nanorod is a pure Al and the outer surface (r = R_{o}) is a pure Si. The properties of the FG nanorod vary through the radius direction based on powerlaw distribution that is stated as [60]

where F(r) = E, , G, , , ; and p is the volume fraction index which is a nonnegative value.
A nanorod with length L and diameter D is considered (Fig. 1). The crosssection of the nanorod is on the xy plane and the origin of the coordinate is set on the left side. The nanorod displacement component for torsional vibration is given as [64, 65]






In the Eqs. (2)(4), is angular displacement about the center of twist, t denotes the time, and , , and are the displacement components in the x, y, and z directions, respectively.
The geometrically nonlinear straindisplacement relationships can be represented using vonKármán theory as follows
, 

, 

, 

, 

, 

. 
where is the strain. After obtaining the strains in the FG nanorod, the stress components in the bulk and the surface of the FG nanorod should be obtained.
The stress components ( ) of the FG nanorod bulk are expressed based on the classical theory of elasticity as follows [66]

Fig. 1. Schematic of FG nanorod. 

(6) 








. 
To obtain the surface stress components, the GurtinMurdoch theory called the surface elasticity theory [66, 67] is used. The constitutive equations of the surface stresses are expressed based on the surface theory as follows

(7) 
where is the surface residual stress under unstrained conditions, and are surface Lamé constants. is the displacement component of the surfaces and δ_{ij} represents the Kronecker delta.
Substituting Eqs.  and into Eq. (7), the surface stress components ( ) for the FG nanorod are obtained as
, 
(8) 
, 

, 

, 

, 

, 
3.2. Governing Equations of Motion
To derive the equations of nonlinear torsional vibration of FG nanorods including the surface energy effect Hamilton’s principle (Eq. (9)) is used.

(9) 
in Eq. (9), T and U are the kinetic and strain energies, respectively, and are given as

(10) 

(11) 
where and are the bulk and surface density, respectively, A is the crosssectional area, V is the volume, and are the bulk and surface kinetic energies, respectively, and and are the bulk and surface strain energies, respectively.
Substituting Eqs. (10) and (11) into Eq. (9) and employing Hamilton's principle result in the nonlinear governing equation of motion for torsional vibration of FG nanorod and corresponding boundary condition as follows

(12) 

(13) 
where, are given as

(14) 









To obtain linear mode shapes and frequencies, the nonlinear parameters in Eqs. (12) and (13) should be ignored. This results in

(15) 

(16) 
The solution of Eq. (15) can be obtained by the separationofvariables method using the following equation

(17) 
where, is the natural linear torsional frequency, is the linear mode shape, and
is the imaginary unit.
Substituting Eq. (17) into Eq. (15) and (16) yields the following equations

(18) 

(19) 
Solving Eq. (18) by considering the conditions given in Eq. (19) results in the FG nanorod’s mode shapes and its natural frequencies with fixed–fixed (fifi) and fixed–free (fifr) boundary conditions as below

(20) 

(21) 

(22) 

(23) 
In the case of nonlinear torsional vibration of FG nanorod, to convert the partial differential equation (Eq. (12)) to an ordinary differential equation, the multimode Galerkin technique (Eq. (24)) is applied.

(24) 
In Eq. (24), denotes a timedependent function to be determined and is the normalized linear mode shape function which can be obtained from Eqs. (20) and (22). Putting Eq. (24) into Eq. (12) yields

(25) 
In the next step, the following dimensionless parameters are defined

(26) 
where q_{max}_{ }denotes the maximum amplitude of the timedependent function .
Using the dimensionless parameters in Eq. (25), multiplying Eq. (25) by the normalized linear mode shape , and integrating from
X = 0 to X = 1, results in the following equation

(27) 
in which the parameters , , and are specified as:

(28) 
and the following relations are used

(29) 
where in Eqs. (28) and (29) and are defined as
·
(fifi end conditions) 
(30) 
·
(fifree end conditions) 
(31) 
The multiple scale method is employed to solve the nonlinear equation, Eq. (27). To this end, the small dimensionless parameter ε is introduced. Therefore, Eq. (27) can be rewritten as follows

(32) 
To use the method of multiple scales, the solution of Eq. (32) can be represented by an expansion having the following form

(33) 
in Eq. (33), t_{0} = t denotes time scale that shows oscillatory effect and .
Substituting Eq. (33) into Eq. (32) and setting the coefficients with a similar power equal to zero leads to the following set of differential equations:

(34) 

(35) 

(36) 
where
and .
A general solution for Eq. (34) can be given as
, . 
(37) 
A_{m} and are a complex function and the complex conjugate of A_{m}, respectively; These functions can be determined by eliminating the secular terms from . For this purpose, and from Eq. (37) should be substituted into Eq. (35).
This results in

(38) 

(39) 
In Eq. (39) CC denotes the complex conjugate of the past terms. The resonance effect in these equations must be eliminated, so we do

(40) 
Now, the solution of Eqs. (38) and (39) are expressed as

(41) 

(42) 
To solve Eq. (40), is expressed in the polar form

(43) 
where a and b are real parameters.
Substituting Eq. (43) into Eq. (40) and separating the real and imaginary sections equal to zero, leads to the following equations

(44) 

where is a constant. Putting the results of Eqs. (44) into Eq. (43), is obtained as

(45) 
Substituting Eqs. (45), (42), (41), and (37) into Eq. (33) results in [69]

(46) 



(47) 
Eqs. (46) and (47) can be rewritten as



(48) 
O( ) 


(49) 
where

(50) 
denotes the nonlinear and is the linear natural frequency. Since represents a bookkeeping device, we put it equal to unity and for satisfying the initial conditions
( ) in Eq. (44), the error related to the secondorder expansion should be taken into account. These yields .
Eq. (49) displays that internal resonances occur in two cases, onetoone , and threetoone .
Before the presentation of graphical and numerical results, a comparative study is conducted to verify the applicability and accuracy of the present formulation. Due to the lack of a similar problem and solution, the accuracy of the present solution is verified by comparing the results with those of Setoodeh et al. [66]. The Young’s modulus and density of the nanorod were taken as 70 GPa and 2700 kg/m^{3}, respectively while the Poisson’s ratio (ϑ) is 0.23. In Table 1 natural frequencies of FGM nanorods are listed for various vibration amplitudes, mode numbers, and nanorod lengths. As shown in Table 1, the reliability of the present formulation and results is confirmed. However, a slight difference between reported frequencies can be observed, which is attributed to different methods of solution. In the current study, the technique of multiple scales has been used to characterize the dynamic characteristics of the nanorod, while Setoodeh et al. [66] used the Homotopy Analysis Method to identify linear and nonlinear torsional free vibration of functionally graded micro/nanotubes.
In the following, the linear and nonlinear frequencies of nanorods are presented by considering the surface effects, the frequency number, the amplitude of nonlinear vibrations, and the radius and the length of the nanorod. In Table 2, the bulk and the surface elastic properties of aluminum (Al) and silicon (Si) are presented. The crystallographic directions of aluminum and silicon are [1 1 1] and [1 0 0], respectively.
The ratio of frequency is calculated as follows to demonstrate how surface components affect the natural torsional frequencies of nanorods:


(51) 
in which is the natural torsional frequency of nanorods with surface energy eﬀects and is the natural torsional frequency of nanorods without surface energy eﬀects. It can be seen from this equation if the frequency ratio is less than one, then the surface parts have a decremental effect and vice versa. Furthermore, as the frequency ratio is one, the natural torsional frequency is not affected by the surface components.
Firstly, the effect of surface energy parameters on the torsional vibration of FG nanorods is investigated in Table 3. Listed in Table 3 are linear and nonlinear torsional frequencies of FG nanorod for various amplitudes of vibration, nanorod radii, mode numbers, and end conditions, i.e. fifi and fifr. Based on this table, the following results can be obtained. In both linear and nonlinear vibrations surface density has a decreasing effect on the natural frequencies of the nanorod. This reduction effect is independent of the amplitude of vibrations, mode number, and type of boundary condition but it depends on the radius of the nanotube and the FG power index. It is seen from Table 3, that as the FG power index grows, the reducing effect of the surface density parameter rises. This is due to the increase in the amount of surface density with increasing the FG power index and since increasing the density increases the kinetic energy, the frequency decreases. Another result obtained from Table 3 is the reduction in the reducing effect of surface density by growing the radius of the nanorod. As another result of Table 3, it can be mentioned the effect of increasing the radius of the nanorod in reducing the effect of surface density.
Table 1. Comparison of nonlinear torsional frequencies (GHz) for various lengths and mode numbers
(all surface parameters considered to be zero).
Amplitude of vibration 
Length (nm) 
Mode number 
Setoodeh et al. [68] 
Present study 
0.001 
10 
1^{st} 
167.422 
167.409 


2^{nd} 
363.744 
363.430 

15 
1^{st} 
109.738 
109.736 


2^{nd} 
228.380 
228.331 
0.010 
10 
1^{st} 
441.133 
434.047 


2^{nd} 
1672.54 
1639.00 

15 
1^{st} 
212.00 
209.533 


2^{nd} 
760.65 
746.79 
Table 2. Bulk and the surface elastic properties of aluminum and silicon
Material 
Bulk elastic properties 
Surface elastic properties 






Al (internal surface) 
28.5 
2700 
10^{7}×5.46 
0.8269 
Si (external surface) 
86 
1250 
10^{7}×3.17 
2.7779 
Table 3. Linear and nonlinear torsional frequency ratios of FG nanorod
Boundary condition 
R_{o }(nm) 
n 

Only 
Only 
Both 

p 
p 
p 

0 
1 
100 
0 
1 
100 
0 
1 
100 

FixedFixed 
1 
1 
0.00 
0.8882 
0.8879 
0.8676 
0.9621 
0.9551 
0.9560 
0.8545 
0.8481 
0.8294 
0.05 
0.8882 
0.8879 
0.8676 
0.9743 
0.9740 
0.9733 
0.8653 
0.8648 
0.8445 

0.10 
0.8882 
0.8879 
0.8676 
0.9896 
0.9937 
0.9933 
0.8789 
0.8824 
0.8618 

5 
0.00 
0.8882 
0.8879 
0.8676 
0.9621 
0.9551 
0.9560 
0.8545 
0.8481 
0.8294 

0.05 
0.8882 
0.8879 
0.8676 
0.9629 
0.9563 
0.9571 
0.8552 
0.8491 
0.8304 

0.10 
0.8882 
0.8879 
0.8676 
0.9650 
0.9597 
0.9601 
0.8571 
0.8521 
0.8330 

2 
1 
0.00 
0.9388 
0.9386 
0.9263 
0.9805 
0.9776 
0.9780 
0.9205 
0.9176 
0.9059 

0.05 
0.9388 
0.9386 
0.9263 
0.9828 
0.9806 
0.9807 
0.9227 
0.9204 
0.9085 

0.10 
0.9388 
0.9386 
0.9263 
0.9877 
0.9867 
0.9865 
0.9272 
0.9262 
0.9138 

5 
0.00 
0.9388 
0.9386 
0.9263 
0.9805 
0.9776 
0.9780 
0.9205 
0.9176 
0.9059 

0.05 
0.9388 
0.9386 
0.9263 
0.9808 
0.9779 
0.9782 
0.9207 
0.9178 
0.9062 

0.10 
0.9388 
0.9386 
0.9263 
0.9814 
0.9786 
0.9790 
0.9213 
0.9185 
0.9068 

FixedFree 
1 
1 
0.00 
0.8882 
0.8879 
0.8676 
0.9621 
0.9551 
0.9560 
0.8545 
0.8481 
0.8294 
0.05 
0.8882 
0.8879 
0.8676 
0.9895 
0.9937 
0.9933 
0.8789 
0.8823 
0.8618 

0.10 
0.8882 
0.8879 
0.8676 
1.0021 
1.0073 
1.0084 
0.8900 
0.8945 
0.8749 

5 
0.00 
0.8882 
0.8879 
0.8676 
0.9621 
0.9551 
0.9560 
0.8545 
0.8481 
0.8294 

0.05 
0.8882 
0.8879 
0.8676 
0.9630 
0.9566 
0.9573 
0.8553 
0.8494 
0.8305 

0.10 
0.8882 
0.8879 
0.8676 
0.9655 
0.9605 
0.9608 
0.8575 
0.8529 
0.8336 

2 
1 
0.00 
0.9388 
0.9386 
0.9263 
0.9805 
0.9776 
0.9780 
0.9205 
0.9176 
0.9059 

0.05 
0.9388 
0.9386 
0.9263 
0.9876 
0.9867 
0.9864 
0.9272 
0.9261 
0.9137 

0.10 
0.9388 
0.9386 
0.9263 
0.9957 
0.9958 
0.9959 
0.9348 
0.9347 
0.9225 

5 
0.00 
0.9388 
0.9386 
0.9263 
0.9805 
0.9776 
0.9780 
0.9205 
0.9176 
0.9059 

0.05 
0.9388 
0.9386 
0.9263 
0.9808 
0.9779 
0.9783 
0.9207 
0.9178 
0.9062 

0.10 
0.9388 
0.9386 
0.9263 
0.9814 
0.9787 
0.9790 
0.9213 
0.9186 
0.9069 
As the radius of the nanorod increases, the energy stored in its volume and surface increases, but the rate of the energy stored in the volume of the nanorod is more than the energy stored on its surface. Therefore, the ratio of the total volume energy to the surface energy will be greater than before. As a result, the effect of surface density becomes less.
According to Table 3, it can be seen that the effect of the Lamé constant on the natural frequency of nanorod is a little different from the effect of the surface density parameter. It is readily apparent from Table 3 that in the case of linear vibrations, the surface Lamé constant has a decreasing effect on the torsional frequencies of the FG nanorod. This decreasing effect is independent of the frequency mode number and boundary end conditions but depends on the value of the power index and the nanorod radius. There is no determined relationship between the reducing effect of the Lamé constant on the natural frequency and the value of the FG power index. As the value of the FG index increases from 0 to 1, the reducing effect of the Lamé constant rises. Then, by growing m from 1 to 100, the decreasing effect of the Lamé constant on the natural frequency reduces. In the end, the reduction effect of the Lamé constant on the natural frequency of nanorod for m = 0 (i.e. Al nanorod) is less than for m = 100 (i.e. Si nanorod). The reason for this is the difference in the amount of Al and Si Lamé constant. Because the dependence of the Lamé constant reduction effect on the FG nanorod radius has a reason similar to the dependence of the surface density reducing effect on the FG nanorod radius, its repetition is avoided.
In general, it can be said that the Lamé constant has a reducing influence on the torsional frequencies of the FG nanorod in the case of nonlinear vibrations. This decreasing effect depends not only on the frequency number and the boundary conditions but also depends on the amplitude of the nonlinear vibrations and the radius of the FG nanorod. However, according to Table 3, for the high amplitude of vibrations and the CF boundary condition, the Lamé constant has an increasing effect. The dependence of the Lamé constant effect on the value of the FG index is similar to that of linear vibrations. Based on Table 3, by increasing the amplitude of nonlinear vibrations and assuming that other parameters do not change, the decreasing effect of the Lamé constant reduces. Because with increasing amplitude of vibrations, the strain energy in the volume of nanorods increases at a higher rate than the strain energy at the surface of the nanorod. The rate of this growth is higher in the higher frequencies number and for the fifi boundary condition. Another result shown in Table 3 is that like linear vibrations, the decreasing effect of the Lame constant decreases with the increasing radius of the FG nanorod. Finally, it can be seen from Table 3 that the reduction effect of the Lamé constant on the torsional frequencies of the FG nanorod is higher for nanorods with fifr end conditions than for nanorods with fifi end conditions.
Finally, the decreasing effects of surface density and surface Lamé constant on the linear and nonlinear torsional frequencies of the FG nanotube become apparent as a cumulative effect when both variables are considered. Table 3 displays that the reducing effect of the effective surface parameters on the torsional vibration behavior of the FG nanorods, in the case of linear vibrations, is independent of the boundary condition and frequency number. However, in the case of nonlinear vibrations, the mentioned reduction effect depends on all factors, including the type of boundary condition, the radius of the nanorod, the amplitude of the vibrations, and the frequency number. The manner of dependence and the type of effect on the nonlinear torsional frequencies of the FG nanotubes are similar to those stated for the case only considering the effect of the Lamé surface constant, and its explanations are omitted to avoid repetition.
The variation of the torsional frequency of the FG nanorod against the nanorod length for various FG power indexes, mode numbers, vibration amplitude, and type of boundary conditions is presented in Figs. 2a2h. It can be found that for shorter nanorods and by increasing the FG power index, the decreasing effect of surface parameters reduces. The reason for the greater decreasing effect of surface parameters on the torsional frequencies of nanorods with shorter lengths is that the shorter the nanorod length, the greater the ratio of energy stored at the surface to energy stored at volume.
In addition, it is also observed that as the length of the FG nanorod increases, both the frequency and the decreasing effect of the surface parameters lessen. This behavior is observed for different values of the FG power index, vibration amplitude, type of boundary condition, and frequency number.
The conditions for occurring internal resonances of CC and CF nanorods with and without the effect of surface parameters are listed in Tables 4a and 4b. It is clear that considering the surface parameters changes the internal resonance ratio, and its value depends on the type of boundary condition. Another point that can be seen from Tables 4a and 4b is that considering the surface parameters and for certain values of m, n, and q_{max}, internal resonances occur for different values of the FG power index at different nanometer lengths. For a given value of m, n, and p by increasing the amplitude of the vibrations, the internal resonances of the nanorod occur in shorter lengths. On the contrary, for certain values of p, n, and q_{max} as the frequency number increases, the internal resonance of the FG nanorod occurs at longer lengths.








Fig. 2. Variations of axial frequencies versus the nanorod length, Ro=2.25 nm, Ri=1.25, a) FiFi, n=1, qmax=0; 
Table 4a. Various conditions for occurring internal resonances of fixedfixed FG nanorod
with and without surface effects (R_{o}=1.5 nm and R_{i}=0.25 nm).
n 
m 
q_{max} 
p 
Length (nm) 


Without surface effect 
With surface effect 

4 
1 
0.07 
0 
20.26 
3.0000 
2.9754 
1 
18.57 
3.0000 
2.9721 

100 
20.16 
3.0000 
2.9715 

4 
1 
0.08 
0 
17.79 
3.0000 
2.9748 
1 
16.31 
3.0000 
2.9718 

100 
17.70 
3.0000 
2.9714 

4 
1 
0.09 
0 
15.88 
3.0000 
2.9754 
1 
14.56 
3.0000 
2.9729 

100 
15.80 
3.0000 
2.9721 

4 
1 
0.10 
0 
14.38 
3.0000 
2.9755 
1 
13.19 
3.0000 
2.9730 

100 
14.31 
3.0000 
2.9718 

5 
1 
0.07 
0 
28.34 
3.0000 
2.9602 
1 
25.97 
3.0000 
2.9556 

100 
28.20 
3.0000 
2.9546 

5 
1 
0.08 
0 
24.85 
3.0000 
2.9596 
1 
22.77 
3.0000 
2.9556 

100 
24.72 
3.0000 
2.8123 

5 
1 
0.09 
0 
22.14 
3.0000 
2.9603 
1 
20.29 
3.0000 
2.9563 

100 
22.03 
3.0000 
2.9548 

5 
1 
0.10 
0 
19.99 
3.0000 
2.9610 
1 
18.33 
3.0000 
2.9562 

100 
19.89 
3.0000 
2.9556 
Table 4b. Various conditions for occurring internal resonances of fixedfree FG nanorod
with and without surface effects (R_{o}=1.5 nm and R_{i}=0.25 nm).
n 
m 
q_{max} 
p 
Length (nm) 


Without surface effect 
With surface effect 

4 
1 
0.07 
0 
19.83 
3.0000 
2.9427 
1 
18.17 
3.0000 
2.9364 

100 
19.72 
3.0000 
2.9369 

4 
1 
0.08 
0 
17.37 
3.0000 
2.9439 
1 
15.92 
3.0000 
2.9370 

100 
17.28 
3.0000 
2.9369 

4 
1 
0.09 
0 
15.47 
3.0000 
2.9440 
1 
14.18 
3.0000 
2.9373 

100 
15.39 
3.0000 
2.9370 

4 
1 
0.10 
0 
13.96 
3.0000 
2.9439 
1 
12.79 
3.0000 
2.9394 

100 
13.89 
3.0000 
2.9364 

5 
1 
0.07 
0 
24.15 
3.0000 
2.9334 
1 
22.13 
3.0000 
2.9257 

100 
24.02 
3.0000 
2.9260 

5 
1 
0.08 
0 
21.26 
3.0000 
2.9153 
1 
19.39 
3.0000 
2.9263 

100 
21.05 
3.0000 
2.9255 

5 
1 
0.09 
0 
18.84 
3.0000 
2.9346 
1 
17.27 
3.0000 
2.9277 

100 
18.74 
3.0000 
2.9269 

5 
1 
0.10 
0 
17.00 
3.0000 
2.9344 
1 
15.58 
3.0000 
2.9279 

100 
16.91 
3.0000 
2.9267 
The variations of the natural frequencies versus vibration amplitude for various cases are presented in Figs. 3a3d. In these figures, the first and fifth modes of FG nanorods with fifi and fifr boundary conditions are considered. It is observed that for both boundary conditions, overall variations of frequencies are the same and this trend can be observed in different mode numbers. In addition, it is revealed that as the amplitude of vibrations increases, the natural frequency of FG nanorod rises and the magnitude of the increase is greater at higher amplitudes. These changes are more obvious at lower mode numbers. The other results are the same as those reported before, therefore they are not repeated here again.
In the end, the effect of radius variations on the natural frequency of FG nanorod is investigated in Figs. 4a4h. By comparing the curves in linear and nonlinear vibration modes, it is observed that the effect of surface energy on linear and nonlinear frequencies is different from each other (as explained before in Table 3). Also, depending on the different mode numbers, the effect of surface energy on the natural frequencies of FG nanorod is various. In Figs 4a4h, the effect of surface energy on the natural frequencies of the nanorod can be seen for different values of the FG power index and different radii of the nanorod (as seen before in Table 3).
In this study, nonlinear torsional vibrations of functionally graded nanorods are investigated using surface elasticity theory and the conditions for which internal resonances occur. Recently, FGMs have been used in micro/nanostructures and atomic force microscopes (AFMs). It is necessary to enhance the knowledge about the mechanical response of the FGMs for the next technological revolution since these structures are emerging as the new generation of micro/nanotubes offering exciting physical and mechanical properties. The main conclusions are as follows:




Fig. 3. Variations of axial frequencies versus the amplitude of vibration, (L=10 nm, Ro=1 nm, Ri=0.5 nm; 








Fig. 4. Variations of frequency versus the nanorod radius (R_{i}=0.25 nm, L=20 nm), a) fixedfixed, n=1, q_{max}=0; 
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript.
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