Document Type : Research Paper
Author
School of Engineering, Damghan University, Damghan, Iran
Abstract
Keywords
Main Subjects
School of Engineering, Damghan University, Damghan, Iran
ARTICLE INFO 

ABSTRACT 
Article history: Received: 20231019 Revised: 20240118 Accepted: 20240315 

In this paper, the size dependency of the nonlinear free longitudinal vibration of axially functionally graded (AFG) nanorods is studied using the nonlocal elasticity theory. A powerlaw distribution is considered for the variations of the nanorod material properties through its length. The nonlinear equation of motion including the vonKarman geometric nonlinearity is derived using Hamilton’s principle. Then, a solution for the linear equation of motion is obtained using the harmonic differential quadrature method, and mode shapes and natural frequencies are extracted. In the next step, the nonlinear natural frequencies are calculated by solving the nonlinear equation of motion using the multiple scales method. Two types of boundary conditions are considered, i.e. fixedfixed and fixedfree. The presented results include effects of various parameters like nanorod length and diameter, amplitude of vibration, small scale parameter, and frequency number, on the natural frequencies. In addition, a comparative study is conducted to evaluate the effects of the type of linear mode shape on the nonlinear natural frequencies. 



Keywords: Nonlinear longitudinal vibration; Nanorod; Functionally graded materials; Nonlocal elasticity theory. 


© 2024 The Author(s). Mechanics of Advanced Composite Structures published by Semnan University Press. This is an open access article under the CCBY 4.0 license. (https://creativecommons.org/licenses/by/4.0/) 
The experimental and theoretical studies on the various behaviors of nanosized structures have shown that they have superior properties in comparison with the structures at the macroscale. These superior properties can be enhanced by using functionally graded materials (FGM) for nanosized structures. FGMs have been used in the nanotechnology field to attain the desired efficiency [14]. Since the behaviors of nanosized FGM structures are different from those made of homogeneous materials, it is incumbent to be intimate with the behaviors of FGM nanostructures for NEMS/MEMS fabrication.
One of the nanoscale elements that attracted the attention of researchers is the nanobeam. There are many works regarding the analysis of its various mechanical behaviors, buckling [5, 6], postbuckling [7, 8], bending [9, 10], stability [11, 12], free and forced axial vibration [13, 14], free and forced torsional vibration [15, 16], free and forced transverse vibration [1720], and wave propagation [21, 22]. Among the studies, the axial behavior of nanobeams (nanorods) has more recently been investigated. These investigations have been done based on three types of sizedependent theories, surface elasticity theory, strain gradient theory, and nonlocal elasticity theory. Free vibration, wave propagation, and instability of nanorods/nanotubes are investigated using the nonlocal elasticity theory. In these researches effects of various factors like elastic medium [2327], magnetic fields [2730], the shape of nanorod cross section (uniform [31, 32], nonuniform [33, 34], tapered [35, 36], conelike [37]), crack [25, 38, 39], discontinuity [40], type of rod theory [26, 4145], number of coupled nanorods [13, 30, 4649], number of nanotube walls [23, 5052], and type of nanorod material properties [5356] have been considered. The other studies in the field of analysis of longitudinal behavior of nanorods/nanotubes are done based on the surface elasticity and strain gradient theories. Free vibration of homogeneous [57, 58], axially functionally graded [5961], viscoelastic [62, 63], and embedded nanorods [64] are analyzed using the strain gradient theory. Finally using the surface elasticity theory, Nazemnezhad and Shokrollahi have considered free longitudinal oscillation of FGM nanosized rods [65].
All of the mentioned works have only analyzed the linear axial behavior of nanorods/nanotubes based on different theories. There are only a few works studying the nonlinear axial vibration of nanotubes/nanorods. Nazemnezhad et al. [66] studied the nonlinear longitudinal oscillation and internal resonances of nanoscale rods incorporating the effects of the surface energy parameters. In another work, the CNTs had been modeled as a homogeneous nanorod, and their nonlinear free axial vibration was investigated [67]. FatahiVajari and Azimzadeh [68] considered the nonlinear axial vibration of singlewalled carbon nanotubes based on the Homotopy perturbation method. In this research, the nonlinear governing equation of motion was derived using doublet mechanics. The nonlinear coupled axial–torsional vibration of singlewalled carbon nanotubes has also been analyzed [69]. In this study, the Galerkin method and homotopy perturbation method were used. In another different work, the internal resonances of nanorods were studied based on the nonlocal elasticity theory [70]. In this work, Nazemnezhad and Zare implemented the multimode Galerkin method to convert the partial differential equation to the ordinary differential type. Jamali Shakhlavi et al. [71] had studied similar research on internal resonances of Rayleigh nanorods. Finally, Jamali Shakhlavi et al. [72] considered the effects of thermal and magnetic fields and elastic medium on the nonlinear axial vibration of nanorods. The nonlocal elasticity was used and the von Kármán geometric nonlinearity was taken into account.
As mentioned in the first paragraph of this section, using functionally graded materials enhances the mechanical properties of structures. In addition, the literature survey showed that there is a lack of investigation of the nonlinear axial vibration of nanotubes/nanorods. The lack is what the effect of material on the nonlinear axial behavior of nanotubes/nanorods is. Therefore, this study presents an investigation of the nonlinear axial behavior of nanorods/nanotubes when they are made of FGMs. To this end, a comprehensive model is proposed to investigate the nonlocal parameter effect on the large amplitude oscillation of nanosize axially functionally graded (AFG) rods with von Kármán type nonlinearity. In this regard, the equation of motion and corresponding boundary conditions are obtained. Then, linear mode shapes and natural frequencies are extracted using the harmonic differential quadrature method. In the next step, using the multiple scales method, the nonlinear governing equation of motion is solved. Finally, the nonlocal parameter effects on the frequencies of AFG nanorods are investigated for various end conditions, nanorod lengths, nanorod diameter, and amplitude ratios.
In order to derive the nonlinear governing equation of motion and corresponding boundary conditions of AFG nanosize rods, an AFG nanosize rod with length L (0 ≤ x ≤ L), crosssectional area A, and diameter D are desired in XYZ coordinates as shown in Figure 1.
Fig. 1. Schematic of the AFG nanorod 
Two different materials at the left and right ends constitute the AFG nanosize rod. The Poisson’s ratio is assumed to be constant but the shear modulus G(x), bulk elastic modulus E(x), and mass density ρ(x) of the AFG nanorod vary in the longitudinal direction according to the power low distribution as follow [73, 74]:



In Eq. 1, m is the gradient index and shows the profile of the mechanical properties of the AFG nanosize rod across its length, and the superscripts R and L imply the right (x=L) and left (x=0) ends of the AFG nanorod, respectively. In Figure 2 variations of the mass density and the Young’s modulus of elasticity across the AFG nanorod length are plotted for various values of the gradient index.
Based on the rod theory, the displacement field at any point of the AFG nanosize rod is given below:


where t is the time, and , and are the displacement components along x, y, and z directions, respectively. To obtain the strain(s) in the AFG nanorod, the vonkármán straindisplacement relation with the nonlinear terms is used

(3) 
Fig. 2. Variations of the mass density and the Young’s modulus across the length of AFG nanorod 
According to Eqs. (3), the nonzero strain in the crosssection of the AFG nanorod is obtained as

(4) 
From Hooke’s law, the stress induced in the crosssection of the AFG nanorod is also obtained as

(5) 
It is now possible to derive the governing equation of motion and the corresponding boundary conditions using Hamilton’s principle given as Eq. (6)

(6) 
where T and U are the kinetic and potential energies of the AFG nanosize rod, respectively, and are given as

(7) 

(8) 
Now substituting Eqs. , (4)(5) into Eqs. (7)(8) results in

(9) 

(10) 
where is the local force resultant and is described as



(11) 
Equations (9) and (10) are the local governing equations of motion and corresponding boundary conditions, respectively.
In order to consider the nonlocal parameter effect on the governing equation of motion of the AFG nanosize rod, the nonlocal elasticity theory [75] is used. To this end, Eq. (9) can be stated in the nonlocal form as

(12) 
where subscript nl denotes the nonlocal, and is the force resultant in nonlocal form or it is the nonlocal force resultant. According to the nonlocal elasticity theory, the can be obtained from Eq. (11) as follows

(13) 
In Eq. (13), is the onedimensional Laplacian operator. Now, is obtained from Eq. (13) as [70]



(14) 
By obtaining the nonlocal force resultant, , the nonlocal governing equation of motion of the AFG nanorod can be given in terms of displacement by substituting Eq. (14) into Eq. (12) as

(15) 
Eq. (15) can be simplified as

(16) 
Eq. (16) can be expanded and rewritten as bellow

(17) 
It is worth noting that by considering Eqs. (10) and (14), the local and nonlocal boundary conditions become the same and are given by

(18) 

Ignoring the nonlinear components in Eqs. (17) and (18) make it possible to analyze the nonlocal linear free longitudinal vibration of AFG nanosize rods. The linear equations are as follows

(19) 

(20) 
Next, it is assumed that the AFG nanosize rod has a harmonic longitudinal displacement as

(21) 
where is the imaginary unit, and is the nonlocal natural axial frequency. By substituting Eq. (21) into Eqs. (19) and (20), the governing equation of motion and the boundary conditions in terms of the displacement are obtained as

(22) 


(23) 

Since the coefficients of Eq. (22) are dependent on x, it is not possible to solve it analytically. To solve Eq. (22) and obtain the natural frequencies and corresponding mode shapes of AFG nanorods, the harmonic differential quadrature method (HDQM) [76], an accurate and useful method for analysis of structural components, is utilized in this study. Consider the domain of structure as which is discretized by N points along
xcoordinate. If represents the deformation function, then its derivative with respect to x at all points can be expressed discretely as

(24) 
in which is the weighting coefficient in conjunction to the jth order derivative of , at the discrete point .
Now, applying Eq. (24) to Eqs. (22)(23) results in the discretized form of the governing equation and boundary conditions as follows

(25) 

(26) 
Arranging the discretized form of Eq. (25) in matrix form, considering the discretized forms of Eq. (26), and separating the domain ( ) and boundary ( ) degrees of freedom (DOF) lead to the following assembled matrix

(27) 
where

(28) 

(29) 


The nonlocal linear longitudinal frequencies, and corresponding mode shapes, can be obtained from Eq. (27) after doing some mathematical simplification as

(30) 
For nonlinear free vibration analysis of nonlocal AFG nanorods, Galerkin’s method is used to convert Eq. (17) to an ordinary differential equation. To this end, the longitudinal displacement of the n^{th} mode of the AFG nanorod can be considered as

(31) 
where U(x) is the nonlocal normalized linear mode shape which can be obtained from Eqs. (27)(30) and T(t) is a time dependent function to be determined. Substituting Eq. (31) into Eq. (17) results in the following ordinary differential equation

(32) 
Then, multiplying Eq. (32) by and integrating from x=0 to x=L results in

(33) 
where

(34) 



Eq. (33) needs two initial conditions as follows

(35) 

(36) 
where T_{max} is a maximum amplitude corresponding to the timedependent function T(t).
The method of multiple scales is implemented to solve Eq. (33). Before that, the following nondimensional quantities are introduced

(37) 

By substituting Eq. (37) into Eqs. (33), (35) and (36), the nondimensional second order ordinary differential equation and initial conditions are obtained as

(38) 

(39) 

(40) 
in which and
.
The method of multiple scales considering a secondorder uniform expansion of the solution of Eq. (38) is desired as

(41) 
where ε is a small parameter measuring the amplitude of oscillation. In addition, we consider a fast time scale, q_{0}=t, denoting the main oscillatory behavior, and slow time scales implying the amplitude and phase modulation.
Since the form of Eq. (38) is the same as the one obtained for nonlinear free transverse vibration of nanobeams [20], the procedure of extracting the nonlocal natural frequency equation of AFG nanorods is not presented here. If we follow the procedure presented in Ref. [20], the nonlocal natural frequency equation of AFG nanorods can be given as

(42) 
where is the nonlocal nonlinear natural frequency of the AFG nanorod, and the subscript and superscript nl denotes the nonlinear and nonlocal, respectively.
Before the presentation of results, it is worth mentioning that for obtaining the nonlinear natural frequencies of AFG nanorods (Eq. (42)), the coefficients defined in Eq. (43) should be calculated. One of the parameters used in the coefficients is the linear normalized mode shapes (Eqs. (28) and (29)) which are obtained in discretized form. Therefore, we do a curvefitting to obtain a function for each linear mode shape. This makes it possible to do mathematical operations on the functions to obtain nonlinear natural frequencies. In Fig. 3, linear mode shapes of AFG nanosize rods for various values of the gradient index are plotted for fixedfixed (fifi) and fixedfree (fifr) end conditions.
In this section, a comparison study is conducted to check the reliability of the present formulation and the analytical solution, and some numerical results are provided for the nonlinear vibration properties of AFG nanosize rods based on the nonlocal elasticity theory.
Table 1 gives a comparison between the natural frequencies of a fifi and a fifr boundary conditions presented by this work with those obtained by Nazemnezhad and Zare [70] for a homogeneous nanoscale rod. Both researchers have used the method of multiple scales to obtain the natural frequencies. In the present study, an AFG nanorod is investigated while in Ref [70] a homogeneous one is desired. Table 1 shows a good agreement in results.
3.2. Benchmark Results
The benchmark results presented here are obtained for nanorods with the mechanical properties given in Table 2.
We first examine the variations of linear and nonlinear nonlocal frequencies versus AFG nanorod diameters. In this regard, the length of the AFG nanorod is equal to 10 (nm) and the corresponding diameter changes are equal to 12 (nm) with a step of 0.25. Since the change in diameter is directly related to the thickness of the AFG nanorod, we have tried to select its values so that the aspect ratio of the AFG nanorod is greater than 10, because in the calculation of nonlinear nonlocal frequencies, the Simple axial beam model has been used, which is suitable for this condition. The results show that with increasing AFG nanorod diameter, the linear nonlocal frequencies are constant, but the nonlinear nonlocal frequencies have increased, although this increase is not significant. Due to the fact that the AFG nanorod material leads to softness, with increasing the gradient index, linear and nonlinear nonlocal frequencies have decreased. Also, due to the fact that the system under study has a hardening effect, with increasing nonlinear amplitude, nonlinear nonlocal frequencies have increased.
The results of these changes are presented in Tables 3 and 4, respectively, for fifi and fifr boundary conditions in terms of the first frequency mode of the AFG nanorod. As the results show, the general trend of changes is the same for both boundary conditions, but with the difference that in the fifi boundary conditions in the first mode, the effect of hardening is maintained, while for the fifr boundary conditions, we see a softening effect in the first mode.
Fig. 3. The first three mode shapes of fixedfixed and fixedfree nanorods for various values of the gradient index. 
Table 1. Comparison study of the present work with Ref. [70] in terms of the first third natural frequencies (GHz)
for AFG nanorod (µ=2 nm, R=2 nm, T_{max} = 2, m=0)
Length (nm) 
Mode number 
fifr BC (E_{R}/E_{L} = 1) 
fifi BC (E_{R}/E_{L} = 1) 

Present work 
Ref. [70] 
Present work 
Ref. [70] 

10 
1 
217.602 
217.602 
453.996 
453.996 
2 
621.698 
621.698 
859.916 
859.916 

3 
1007.970 
1007.970 
1270.760 
1270.760 

20 
1 
115.402 
115.402 
232.917 
232.917 
2 
339.767 
339.767 
453.996 
453.996 

3 
550.372 
550.372 
660.864 
660.864 
Table 2. Mechanical properties of AFG nanorod
Parameter 





value 
70 
210 
2700 
2370 
0.3 
Table 3. Variations of longitudinal frequencies versus diameters of AFG nanorod in terms of 

D 
m = 0 
m = 1 
m = 10 

µ = 0 
µ = 1 
µ = 2 
µ = 0 
µ = 1 
µ = 2 
µ = 0 
µ = 1 
µ = 2 

Nonlinear amplitude (T_{max} = 0) 

1 
234.806 
231.974 
229.242 
178.002 
175.633 
173.344 
139.397 
137.577 
135.823 
1.25 
234.806 
231.974 
229.242 
178.002 
175.633 
173.344 
139.397 
137.577 
135.823 
1.5 
234.806 
231.974 
229.242 
178.002 
175.633 
173.344 
139.397 
137.577 
135.823 
1.75 
234.806 
231.974 
229.242 
178.002 
175.633 
173.344 
139.397 
137.577 
135.823 
2 
234.806 
231.974 
229.242 
178.002 
175.633 
173.344 
139.397 
137.577 
135.823 
Nonlinear amplitude (T_{max} = 1) 

1 
235.010 
232.176 
229.441 
178.161 
175.787 
173.494 
139.503 
137.681 
135.926 
1.25 
235.125 
232.289 
229.553 
178.250 
175.874 
173.578 
139.562 
137.739 
135.983 
1.5 
235.265 
232.427 
229.690 
178.359 
175.980 
173.681 
139.635 
137.810 
136.053 
1.75 
235.430 
232.591 
229.852 
178.488 
176.106 
173.803 
139.720 
137.895 
136.136 
2 
235.621 
232.780 
230.038 
178.637 
176.251 
173.943 
139.819 
137.992 
136.232 
Nonlinear amplitude (T_{max} = 2) 

1 
235.621 
232.780 
230.038 
178.637 
176.251 
173.943 
139.819 
137.992 
136.232 
1.25 
236.080 
233.233 
230.486 
178.995 
176.598 
174.280 
140.057 
138.226 
136.462 
1.5 
236.641 
233.787 
231.033 
179.432 
177.023 
174.692 
140.347 
138.512 
135.926 
1.75 
237.303 
234.441 
231.680 
179.948 
177.525 
175.179 
140.691 
138.849 
136.743 
2 
238.068 
235.196 
232.427 
180.544 
178.104 
175.741 
141.087 
139.239 
137.458 
Table 4. Variations of longitudinal frequencies versus diameters of AFG nanorod in terms of 

D 
m = 0 
m = 1 
m = 10 

µ = 0 
µ = 1 
µ = 2 
µ = 0 
µ = 1 
µ = 2 
µ = 0 
µ = 1 
µ = 2 

Nonlinear amplitude (T_{max} = 0) 

1 
117.627 
117.266 
116.908 
98.468 
98.145 
97.825 
70.243 
70.008 
69.7747 
1.25 
117.627 
117.266 
116.908 
98.468 
98.145 
97.825 
70.243 
70.008 
69.7747 
1.5 
117.627 
117.266 
116.908 
98.468 
98.145 
97.825 
70.243 
70.008 
69.7747 
1.75 
117.627 
117.266 
116.908 
98.468 
98.145 
97.825 
70.243 
70.008 
69.7747 
2 
117.627 
117.266 
116.908 
98.468 
98.145 
97.825 
70.243 
70.008 
69.7747 
Nonlinear amplitude (T_{max} = 1) 

1 
117.530 
117.169 
116.812 
98.408 
98.085 
97.765 
70.191 
69.956 
69.723 
1.25 
117.475 
117.115 
116.758 
98.374 
98.051 
97.731 
70.162 
69.927 
69.694 
1.5 
117.408 
117.048 
116.691 
98.332 
98.010 
97.690 
70.126 
69.891 
69.658 
1.75 
117.330 
116.970 
116.613 
98.283 
97.961 
97.642 
70.084 
69.849 
69.616 
2 
117.239 
116.879 
116.523 
98.227 
97.905 
97.586 
70.035 
69.801 
69.568 
Nonlinear amplitude (T_{max} = 2) 

1 
117.239 
116.879 
116.523 
98.227 
97.905 
97.586 
70.035 
69.801 
69.568 
1.25 
117.021 
116.662 
116.306 
98.091 
97.769 
97.451 
69.919 
69.684 
69.452 
1.5 
116.754 
116.396 
116.041 
97.924 
97.604 
97.287 
69.776 
69.542 
69.310 
1.75 
116.439 
116.081 
115.727 
97.728 
97.405 
97.092 
69.607 
69.374 
69.143 
2 
116.075 
115.719 
115.366 
97.501 
97.183 
96.868 
69.413 
69.180 
68.949 
Next, the linear and nonlinear frequency changes of the AFG nanorod are studied based on the variations of the nonlocal parameter and the gradient index for the first and second AFG nanorod modes in terms of both fifi and fifr boundary conditions. These results are derived for AFG nanorods with a length of 10 (nm) and a diameter of 1 (nm) in terms of three nonlinear amplitudes of 0, 1 and 2. The results show that the decreasing effects of nonlocal parameters on linear and nonlinear frequencies for the first mode are negligible. Also, the decreasing slope of the graphs for the lower gradient index is higher, which indicates that the stricter the AFG nanorod, the greater the decreasing effects of the nonlocal parameter on its linear and nonlinear frequencies. Here, too, the trend of general results changes for both fifi and fifr boundary conditions is similar, except that the frequencyreducing effects of the nonlocal parameter for
fifr boundary conditions are less than fifi. The detailed results of the study of these changes can be clearly seen in Figs 4a to 4f.
In this section, the longitudinal frequency changes of AFG nanorods based on nonlinear amplitudes for different nonlocal parameter values and gradient indices are evaluated. In this evaluation, the length and diameter of the AFG nanorod are considered equal to 10 and 1 (nm), respectively. As shown in Fig 5, the variations of the longitudinal frequencies of the AFG nanorod versus the nonlinear amplitude changes are greater for the fifi than fifr boundary conditions. Also, the incremental effect of nonlinear nonlocal frequencies due to the increase of nonlinear amplitude for the second mode of AFG nanorod frequency is higher than the first mode, but this incremental effect will be more in higher modes (third mode onwards). In addition, the results show that the incremental effects of increasing the nonlinear amplitude are the same for different gradient index nanorods. In other words, the nonlinear effects of material changes along the AFG nanorod are neutral. The results of these studies are clearly shown in the various sections of Fig 5a, 5b, 5c, and 5d for both boundary conditions.
Finally, the frequency changes of the AFG nanorod versus its length for different nonlocal parameter values and gradient index are presented in Fig 6. These results are derived for both fifi (Figs. 6a, 6b, and 6c) and fifr (Figs. 6d, 6e, and 6f) boundary conditions for an AFG nanorod with nonlinear amplitudes of 0, 1, and 2. Figs 6a, 6b, and 6c show that the decreasing effects of linear and nonlinear frequencies due to the nonlocal parameter in the fifi boundary conditions are greater than the fifr, which has been observed in previous results. Another important point is that with increasing AFG nanorod length, linear and nonlinear nonlocal frequencies coincide, which shows that the effects of changes in material, nonlocal parameter, and nonlinear amplitude on long AFG nanorods on its frequencies are negligible. The bottom line is that the larger the AFG nanorod length, the higher the nonlinear and nonlocal coefficients must be for the results to be significant. Also on thick nanorods, the effects of nonlinear factors are less than on thin nanorods, see Fig. 6.
a) fifi, T_{max} = 0 
d) fifr, T_{max} = 0 
b) fifi, T_{max} = 1 
e) fifr, T_{max} = 1 
c) fifi, _{Tmax} = 2 
f) fi_{fr}, T_{max} = 2 
Fig. 4. Variations of longitudinal frequencies versus nonlocal parameters for various gradient indexes 
In the present study, large amplitude free longitudinal vibration behavior of axially functionally graded nanorods with material gradation is investigated. Different boundary conditions are considered namely fifi and fifr. However, the present methodology can be employed for other types of classical and nonclassical boundaries as well. Hamilton’s principle is utilized for the mathematical formulation and the boundary conditions, the problem can be solved in two parts, linear and nonlinear with the help of Harmonic Differential Quadrature and Multiple Scale methods, respectively. The methodology is general in nature as it can be applied to another type of material gradation and rod patterns. The obtained results are confirmed from previously published results and were found to be in good agreement. Results pertaining to various AFG nanorod parameters for fifi and fifr boundary conditions are furnished as highlight curves for the fundamental modes. For all combinations of the system parameters, the hardening type of nonlinearity is observed in terms of fifi boundary conditions while the softening type is demonstrated for initial modes in terms of fifr boundary conditions. Nonlinear nonlocal analysis of rodtype structural components of NEMS (NanoElectroMechanicalSystems) is a topic of major interest in Engineering and Medical sciences.
a) fifi, n=1 
b) fifr, n=1 

c) fifi, n=2 
d) fifr, n=2 

Fig. 5. Variations of longitudinal frequencies versus nonlinear amplitudes for various nonlocal parameters 

a) fifi, T_{max} = 0 
d) fifr, T_{max} = 0 

b) fifi, T_{max} = 1 
e) fifr, T_{max} = 1 

c) fifi, T_{max} = 2 
f) fifr, T_{max} = 2 

Fig. 6. Variations of longitudinal frequencies versus length of AFG nanorod for various nonlocal parameters 

The financial support of the research council of Damghan University with grant number 97/Engi/133/307 is acknowledged.
The author declares that there is no conflict of interest regarding the publication of this article.
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