J. Cent. South Univ. (2018) 25: 461-472

DOI: https://doi.org/10.1007/s11771-018-3750-7

Parametric modeling of carbon nanotubes and estimating nonlocal constant using simulated vibration signals-ARMA and ANN based approach

Saeed Lotfan, Reza Fathi

Young Researchers and Elite Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: Nonlocal continuum mechanics is a popular growing theory for investigating the dynamic behavior of Carbon nanotubes (CNTs). Estimating the nonlocal constant is a crucial step in mathematical modeling of CNTs vibration behavior based on this theory. Accordingly, in this study a vibration-based nonlocal parameter estimation technique, which can be competitive because of its lower instrumentation and data analysis costs, is proposed. To this end, the nonlocal models of the CNT by using the linear and nonlinear theories are established. Then, time response of the CNT to impulsive force is derived by solving the governing equations numerically. By using these time responses the parametric model of the CNT is constructed via the autoregressive moving average (ARMA) method. The appropriate ARMA parameters, which are chosen by an introduced feature reduction technique, are considered features to identify the value of the nonlocal constant. In this regard, a multi-layer perceptron (MLP) network has been trained to construct the complex relation between the ARMA parameters and the nonlocal constant. After training the MLP, based on the assumed linear and nonlinear models, the ability of the proposed method is evaluated and it is shown that the nonlocal parameter can be estimated with high accuracy in the presence/absence of nonlinearity.

Key words: nonlocal theory; nonlocal parameter estimation; autoregressive moving average; artificial neural network; feature reduction

Cite this article as: Saeed Lotfan, Reza Fathi. Parametric modeling of carbon nanotubes and estimating nonlocal constant using simulated vibration signals-ARMA and ANN based approach [J]. Journal of Central South University, 2018, 25(3): 461–472. DOI: https://doi.org/10.1007/s11771-018-3750-7.

1 Introduction

An understanding of the dynamic behavior of the carbon nanotubes (CNTs) is very important for exploring the applications in nanoscale systems. To this end, a large number of investigations have been done to study the dynamic behavior of CNTs. The classic beam theory can be used to predict the key parameters of the vibration behavior [1, 2]. However, to simulate the dynamic properties of CNTs more carefully one may use molecular dynamics (MDs) simulations or experimental approach [3]. For example, CAO and CHEN [4] studied the bending buckling behavior of single- walled carbon nanotubes (SWCNTs) by implementing both molecular dynamics simulation and finite element method. The comparison between results of the two methods revealed that the continuum shell model cannot quantitatively reproduce the bending buckling behavior of SWCNTs. SRIVASTAVA and BARNARD [5] investigated the axial buckling of single and multiwall nanotubes by using molecular dynamics simulation method. TREACY et al [6] estimated the elastic modulus of isolated CNTs by measuring the amplitude of their intrinsic thermal vibrations via the transmission electron microscope. They observed that carbon nanotubes have exceptionally high elastic moduli, in the terapascal range. It is important to note that the controlled experiment at nanoscale is difficult and atomistic computational methods such as molecular dynamic simulations are computationally expensive and complex for nanostructures with large-sized atomic systems [7–10]. In this regard, implementing the nonlocal theory, which is a reliable and low cost method, may be used for modeling and studying the vibration behavior of CNTs [11, 12]. Nonlocal elasticity theory considers the scale effects by assuming the stress at a point to be a function of strain field at every other point in the structure. ANSARI et al [13] developed a nonlocal continuum shell model by incorporating of Eringen’s nonlocal elasticity equations into the classical Donnell shell theory to study the free vibration response of double-walled carbon nanotubes (DWCNTs). They investigated the vibration behavior of DWCNTs by using nonlocal continuum shell model and compared the obtained results with those of MD simulations. They concluded that with choosing proper values of nonlocal parameter, the proposed theory has a good capability to predict the vibration behavior of DWCNTs. REZAEE and LOTFAN [14] studied the transverse vibration of axially moving CNTs with time dependent velocity based on nonlocal theory. They showed that by increasing the nonlocal parameter, the critical axial velocity reduces. AYDOGDU [15] used a nonlocal elastic rod model to investigate the small scale effect on the axial vibration of nanorods. He observed that the small scale constant has a major role on the values of axial vibration frequencies. ANSARI et al [16] investigated the free vibration of a fractional viscoelastic Timoshenko nanobeam based on nonlocal elasticity theory. The results showed that the vibration frequency of the system increases by increasing the viscoelasticity coefficient. PRADHAN and MANDAL [17] studied the thermal effect on response of CNTs using nonlocal elasticity and Timoshenko beam theories. They found from the results that the ratios such as frequency, buckling, and deflection are decreasing, decreasing and increasing, respectively, with the increase in nonlocal constant. However, the mentioned parameters are increasing, increasing and decreasing, with the increase in temperature change, respectively. HOSEINZADEH and KHADEM [18] investigated thermoelastic vibration behavior of a double-walled carbon nanotube by using nonlocal shell theory. They modeled the DWCNTs as two individual cylindrical thin shells. The obtained results show that the small scale effects increase thermoelastic damping and decrease natural frequencies compared to the local model. MURMU and PRADHAN [19] implemented nonlocal elasticity and Timoshenko beam theory to investigate the buckling of a SWCNTs embedded in an elastic medium. They observed that the critical buckling loads of SWCNT are strongly dependent on the nonlocal small scale parameter and on the stiffness of the surrounding medium.

By considering above researches, one can obtained that the small scale parameter is a key constant to model the vibration behavior of CNTs in nonlocal theory and should be estimated accurately. To this end, here a new method is proposed to estimate the nonlocal parameter using the time response of the CNT. The parametric model of the CNT is developed by autoregressive moving average (ARMA) method for the first time and the estimation of the nonlocal constant is carried out based on this model and implementing artificial neural network (ANN). The remaining sections of the paper are organized as follow. In Section 2, the mathematical model of the transverse vibration of the CNT under impulsive force is developed based on nonlocal theory. By considering the large deformations in the system, the nonlinearity is taken into account. In Section 3, the parametric model of the CNT based on ARMA method is constructed and the ARMA parameters are calculated. In Section 4, two neural networks based on linear and nonlinear models are trained to estimate the nonlocal constant. It should be mentioned that a new feature reduction technique is introduced to select ARMA parameters that are more sensitive to the nonlocal constant. These selected parameters are considered as input vectors of the ANNs. Finally, in Section 5 the results are discussed in detail.

2 Mathematical model of CNT

A single-walled carbon nanotube, pinned at both ends is considered as shown in Figure 1. The CNT has elastic modulus E, cross-sectional area moment of inertia I, length L, inner radius r₁ and

thickness h. In addition, shows the external impulsive excitation exerted on the midpoint of the CNT, and is the planar motion of the CNT along the axis at location and time

Figure 1 Schematic of a pinned-pinned CNT under impulsive force

In this study the system is excited by the impulsive force and the response of an arbitrary point along the CNT is estimated. In other words, the CNT is simulated as a single-input-single-output (SISO) system.

2.1 Equation of motion

In this section, by using the nonlocal beam theory and implementing Hamilton’s principle, the governing equation of motion is obtained. First, the kinetic energy of the system is expressed as follows:

                    (1)

where ρA_nt is the mass per unit length of the nanotube. By assuming the possibility of exerting high impulsive force or equivalent initial displacement on the CNT, the geometric nonlinearity due to the large deformation is taken into account and the Lagrangian strain is used as follows [20]:

                      (2)

The nonlocal beam theory is used to consider the small scale effects. According to this theory, the stress at a point of a body is dependent on the strain at all the points in the body. For a homogenous isotropic material, ERINGEN [21] proposed the following equation as the nonlocal stress field:

                     (3)

where is the local axial stress; is the nonlocal axial stress; e₀ is the material constant determined experimentally and a is the characteristic length. The parameter e₀a is a function of the boundary conditions and molecular lattice [22, 23]. The bending moment and axial force of the CNT due to the vibration are defined respectively as follows:

                        (4)

                          (5)

in the above equations, A_nt denotes the cross- sectional area of the CNT. The strain energy of the CNT is also given by

       (6)

Using Hamilton's principle for the non- conservative system, the following forced equation of motion can be derived:

(7)

where δ is the Dirac delta function. By ignoring the longitudinal vibration, the following equation can be expressed for the axial force:

                                (8)

By using Eqs. (2) to (4) and Eqs. (7) and (8), the governing equation of motion under impulsive force can be derived as the following form:

   (9)

Defining the following dimensionless parameters:

                                 (10)

                                   (11)

                            (12)

                                  (13)

                                (14)

                           (15)

the dimensionless governing equation of motion is obtained as follows:

     (16)

In Eqs. (10) to (16), w, x, t, τ, γ and F are dimensionless transverse displacement, axial coordinate, time, nonlocal parameter, nonlinear parameter, and excitation, respectively. Corresponding boundary conditions for solving Eq. (16) for the case of both ends pinned are also given by

                       (17)

                   (18)

The nonlinearity in Eq. (16) is due to the large deformation of the CNT, and the nonlocal terms in this equation appear in both linear and nonlinear parts. In addition, the impulsive force is derived as a nonlocal force in the system which is exerted on the midpoint of the CNT.

2.2 Approximate solution

Here, the equation of motion is discretized based on the approach utilized by LEE and CHUNG [24]. Accordingly, the solution is expressed as a series of basis functions in the following form:

                    (19)

where N is the total number of basis functions; W_i(x) is the basis function; and T_i(t) is an unknown function of time to be determined. According to the boundary conditions of the CNT, the comparison function is assumed as

               (20)

in which b_i is a constant. The weighting functions corresponding to the trial functions are also given by

                    (21)

where is an arbitrary function of time. Substituting w from Eq. (19) into Eq. (16), multiplying this equation by from Eq. (21), summing all the equations, integrating them over the length of the CNT, and then collecting all the terms with respect to provide the discretized equations in the following form:

                       (22)

in which, local parameters are given by

                  (23a)

                  (23b)

    (23c)

and nonlocal parameters are

                  (24a)

   (24b)

    (24c)

   (24d)

       (24e)

By assuming an appropriate value for N and solving Eq. (22) numerically, the approximate solution can be obtained using Eqs. (19) and (20).

2.3 Simulation results

In order to derive the time response of the CNT to the impulsive force, the nonlinear equation of motion Eq. (22) is solved by using a 4th order Runge–kutta method. The CNT has elastic modulus E=1.1 TPa, length L=45 nm, inner radius r₁=0.34 nm, effective thickness h=0.34 nm, and mass density ρ=1300 kg/m³. The time responses of the CNT to the impulsive force based on linear and nonlinear models are depicted in Figure 2 for the case of τ = 0.

Figure 2 Time response of an arbitrary point along CNT, i.e. x=1/7 based on linear and nonlinear models forτ=0

For validation of the numerical solutions, the above time responses are transferred to the frequency domain to check the values of the resonance frequencies. Corresponding power spectral densities in the range of the first four natural frequencies are depicted in Figure 3. As seen from this figure the nonlinearity increases the natural frequencies of the system and regardless of nonlinearity, the resonance frequencies based on the local theory (τ=0) are close to the exact natural frequencies of the system, i.e. (nπ)².

Figure 3 Power spectral density in range of the first four resonance frequencies of system for linear and nonlinear models for τ=0

Once the mathematical model is developed, the parametric black box model of the CNT is established in the next section.

3 Parametric model of CNT

The identification procedure consists of finding mathematical models of dynamical systems based on observed data. The mathematical model resulting from system identification can be further used to carry out simulation and prediction, which involves a wide range of applications such as mechanical engineering, economics and control design. It is important to note that one of the most crucial identification steps depends on the assumptions that are made on the model structure. In engineering science, for a system with known input and output data, a black box as shown in Figure 4, can be used to build the structure of the model and estimate the parameters of the system.

Figure 4 Schematic of a black box model

In this work by considering the black box model and implementing the autoregressive moving average method, the CNT is simulated. For this purpose, first the method of system modeling by ARMA is described in the following.

3.1 ARMA method: general considerations

ARMA(n, m) models consist of two parts, the autoregressive (AR) part and the moving average (MA) part, where n is the order of the autoregressive part and m is the order of the moving average part. In this model for simulation of the dynamic system, the ARMA parameters which depend on the nature of the system should be estimated. To build the ARMA model, the signal of the system response is considered by ARMA(n, m) as follows:

           (25)

where t, t₀, w_t and e_t are discrete time, starting time, signal for modeling, and white noise with variance respectively; a_i and c_i are parameters of autoregressive and moving average parts of ARMA model. Let B represent the backshift operator:

                        (26)

Consider the ARMA model as:

                (27)

The polynomial defined in Eq. (27) can be manipulated by defining theas:

                    (28)

So, the model in Eq. (25) can be rewritten as follows:

                             (29)

where I[B] is:

                    (30)

with

                       (31)

The algorithm for estimation of the model parameters is summarized as below [15]:

1) First orders of the system i.e., n and m should be defined and estimate parameters of I[B] by appropriate truncation in Eq. (29) by linear least square method.

2) Obtain the initial estimate of MA parameters by using Eq. (30).

3) Determine AR parameters corresponding to initial MA parameters by Eq. (31).

4) Determine polynomial operator β(B) from:

        (32)

5) Filter the signal w_t through β(B).

6) Determine the new AR coefficient by solving the linear least square equation:

                            (33)

where is the filtered signal. Considering that the estimation process is done correctly, the residual ε_t would be an accurate approximate of e_t in Eq. (25).

7) Update MA parameters using Eq. (30).

It is to note that steps 4 to 7 may be repeated to reach an acceptable convergence to estimate model parameters.

3.2 ARMA model results

The simulated time responses from x=1/7 are considered as basis signal for the system identification. According to the above descriptions, in the case of modeling the signal, the order of the system should be defined. For this purpose, Akaike Information Criterion (AIC) method (see Ref. [25]) is used in this paper to obtain the proper order of the model. The investigations of various cases prove that ARMA(6,6) and ARMA(7,7) are the best orders for estimation of this system based on linear and nonlinear models, respectively.

In Figures 5 and 6 the time responses of the CNT for the linear and nonlinear models and their correspondence estimated signals based on ARMA(6,6) and ARMA(7,7) are depicted, respectively. As seen from these figures, ARMA(6,6) and ARMA(7,7) models can estimate both of the signals perfectly.

Figure 5 Simulated and estimated time response of CNT based on the linear theory for τ=0.01.

Figure 6 Simulated and estimated time response of CNT based on nonlinear theory for τ=0.1

4 Estimation of nonlocal parameter

In this section, the nonlocal parameter, τ, is estimated based on the ARMA model using artificial neural networks. The first and major step in estimating this value is extracting appropriate features. The features which solve the inverse or identifications problems in structural dynamics can be obtained by modal characteristics [26]. One of the most informative features in identifying system parameters is ARMA model which have a higher heritability of the physical nature of the structure [27, 28]. Here, the autoregressive parameters in the ARMA model are used as the features. To this end, 61 equally distributed values of nonlocal parameter in the range 0<τ<0.6 are used and correspondence autoregressive parameters are obtained. As discussed above there are six and seven autoregressive parameters for the linear and nonlinear models, respectively; however, in order to reduce the number of theses parameters to a lower number of sensitive parameters, a feature reduction technique is applied in the following.

4.1 Feature reduction

Consideration of six or seven ARMA parameters as the input vector for the neural network and assuming 61 classes may make the process of constructing the neural network difficult and time consuming. In this regard, here the effect of nonlocal constant on the mentioned parameters is analyzed and parameters which are more sensitive to this constant are considered as inputs of the neural network. In order to determine these sensitive parameters, the following steps should be taken:

1) The time response of the CNT for each nonlocal parameter is derived.

2) ARMA parameters are estimated for each signal.

3) As shown in Figure 7, the absolute distance, d_i (i=1, 2, …, 60), between the first parameter of the first class and the first parameter of other 60 classes, are calculated.

4) Step 3 is repeated 60 times and in each time, one class is considered as the basis and the absolute distance between the first parameter of other classes and the base class is computed.

5) The sum of computed distances for the first parameter for various described conditions is considered as the distribution factor for the first parameter.

6) Distribution factors for the other five parameters are calculated as in Steps 3, 4 and 5.

7) The parameters which have higher distribution factor are considered as sensitive parameters to nonlocal constant and are assumed as inputs of the neural network.

Figure 7 Calculation of absolute distance between first parameter of first class and first parameter of other classes

In Tables 1 and 2, the normalized distribution factor of ARMA parameters for linear and nonlinear models of the CNT are presented, respectively. By assuming d_i>0.1 as the sensitivity criterion, parameter a₆, for the linear case and parameters a₆ and a₇ for the nonlinear case are omitted from the input vectors of the neural network.

Table 1 Sensitivity index of autoregressive parameters to nonlocal constant based on linear model of CNT

Table 2 Sensitivity index of autoregressive parameters to nonlocal constant based on nonlinear model of CNT

4.2 ANN structure

A multi-layer perceptron which is capable of modelling complex relationships between variables is established to build up the estimator. The structure of the MLP is a layered feed-forward network, in which the neurons are organized in successive layers, and the information flows in a unidirectional way from the input layer to the hidden layer(s) and then to the output layer. Normally, the sth neuron in a layer of an MLP has an input vector of n individuals, p₁, p₂, …, p_n, each of which is weighted by corresponding elements w_s,_i (i=1, 2, …, n). The bias b_s is then added to the weighted inputs to form the net input, u_s, and finally the sth neuron output, f(u_s), is calculated by using the transfer function f [29]. The transfer function may be a linear or nonlinear function of the net input, for instance hardlim, satlins, logsig, etc. However, it has been proved that the multilayer feed-forward networks, under very general conditions of the hidden layer transfer function, are universal approximators, providing that sufficiently enough hidden layers are available [30].

Experimental applications in several researches have also showed that the multi-layer perceptron having one hidden layer is one of the simplest and most effective networks, which can be used in many complex systems because it includes few controlling parameters to be adjusted. The number of neurons used in the input and output layers, correspond to the number of input and output parameters, respectively. So, the network includes five neurons in the input layer and one neuron in the output layer. The simple architecture of the used ANN is depicted in Figure 8.

The number of neurons in the hidden layer is significant for the performance of the ANN. In this study, an optimum number of neurons is determined based on the minimum value of mean square error (MSE) in the network [31]. The logsig function is employed for both the input and hidden layers to create a smooth convergence. This function has been usually used because it is differentiable, continuous and nonlinear function which provides a self-limiting behavior in a way that its output cannot grow considerably large or small [32, 33]. For the output layer purelin function is employed to map the output values to any essential range. The transfer function of the ith layer is given by

Figure 8 Simple architecture of developed ANN

                   (34)

The detailed structure of the used ANN in abbreviated notation is shown in Figure 9. In this figure, P is the input vector of the ANN, Wⁱ, bⁱ, uⁱ, and vⁱ are the ith layer’s weight, bias, net input and output matrices, respectively, and s denotes the number of neurons in the hidden layer.

4.3 Training and testing ANN

For training the ANN based on the simulated data, a set of input-target pattern pairs is used. There are 61 classes in this study, 43 of which are chosen randomly and used for training the neural network, 9 classes for the network’s cross- validation (to avoid over-fitting) and the remaining 9 classes are used for testing the performance of the trained network. The training process is based on the backpropagation (BP) learning scheme which is an error minimization algorithm based on gradient decent approach to optimize the network performance and obtain the optimum values of the weights and biases [29]. There are several backpropagation-based training strategies such as Levenberg-Marquardt, Bayesian regulation, conjugate gradient, which provide faster solutions because of the incorporation of an extra second derivative of error and automatic internal modifications that are made to the learning parameters [34]. In this study, Levenberg- Marquardt backpropagation algorithm which is popular and more efficient is implemented.

Figure 9 Detailed structure of developed ANN

As mentioned before, the optimum number of neurons in the hidden layer is obtained based on the minimum value of MSE. For this purpose, the ANN is trained by using varying neuron number in the range of 1–20 to obtain an optimum value for s.Figure 10 shows the relationship between the number of neurons and MSE for prediction of nonlocal parameter based on the linear and nonlinear models, respectively.For one neuron, the value of mean square error based on linear and nonlinear models are 2.80×10^–4, and 1.63×10^–4, respectively. These values decrease and reach optimum values as the number of neurons increases. According to this figure, the number of neurons in the hidden layer of the ANN is chosen to be 8. Therefore, the architecture of the used ANN is 5-8-1.

Figure 10 Relationship between number of neurons in hidden layer and MSE

The performance of the ANN should be tested by four standard criteria including the correlation coefficient, R, mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE). These criteria are respectively given by [35]:

                    (35)

                       (36)

                 (37)

                     (38)

where t is the target; o is the ANN output and C is the number of classes. The goal is to maximize correlation coefficient and minimize the error parameters to obtain a network with the best generalization. MAE is a linear score that measures the average error in a set of forecasts and all the individual differences are weighted equally. RMSE is a quadratic scoring rule which measures the average squared error and gives a relatively high weight to large errors. This means that RMSE is most useful when large errors are particularly undesirable. It should be mentioned that both values of MAE and RMSE are absolute and their values may be in any range based on the range of the target values. However, MAPE is a better measure of accuracy in prediction results since its relativity provides general criteria on for performance of the ANN. MAPE value of 20% may be a limit for the performance of an ANN [36, 37]. The results of training the ANN estimating the nonlocal parameter are depicted in Figures 11 and 12 for the linear and nonlinear models, respectively. The comparison between network outputs and targets for the train, test and validation data are shown in these figures.

For evaluating the performance of the trained ANN more carefully, the four aforementioned standard criteria are given in Table 3. As discussed above, MAPE value is a more general criterion in evaluating the ANN accuracy. According to the results, this criterion is less than the limit 0.2 for all data, which indicates a good performance of the ANN.

Figure 11 Comparison between estimated nonlocal parameter and target values for train, test and validation data based on linear theory

Figure 12 Comparison between estimated nonlocal parameter and target values for train, test and validation data based on nonlinear theory

Table 3 Four standard performance criteria based on trained ANN

It should be mentioned that the parametric modelling and the subsequent estimation of the nonlocal constant can be applied based on MDs or experimental results; however, in this research to evaluate the feasibility of the proposed approach the simulated nonlocal model is used instead.

5 Conclusions

In this study, the parametric ARMA model of the CNT’s transverse vibration response to an impulsive force is constructed based on mathematical nonlocal linear and nonlinear simulations. Autoregressive parameters of the model, after feature reduction, are used as inputs of the neural network to establish the complex relation between the nonlocal constant and these features. The main results of this investigation are as following:

1) ARMA method can perfectly simulate the CNT for linear and nonlinear models.

2) The trained networks can estimate the value of nonlocal constant based on autoregressive parameters with high accuracy. The average correlation factors for estimation of the nonlocal parameter for the linear and nonlinear models are 0.9989 and 0.9999, respectively.

3) The interesting result is that the proposed methodology is robust to the presence of nonlinearity in the system and even the average MAPE value for the nonlinear case, i.e. 0.0436, is less than that of the linear case, i.e. 0.0726. This is because of the fact that the nonlocal nonlinear terms appear in the model which can increase the effect of nonlocality on the system response.

4) The estimation of the nonlocal parameter by using the proposed approach (i.e. ARMA and ANN based methodology) is a straightforward and easy method which can be implemented based on mathematical simulation (nonlocal theory), MDs, and experimental results.

References

[1] WANG L, NI Q, LI M, QIAN Q. The thermal effect on vibration and instability of carbon nanotubes conveying fluid [J]. Physica E: Low-dimensional Systems and Nanostructures, 2008, 40(10): 3179–3182.

[2] REZAEE M, MALEKI V A. An analytical solution for vibration analysis of carbon nanotube conveying viscose fluid embedded in visco-elastic medium [J]. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2015, 229(4): 644–650.

[3] FATHI R, LOTFAN S, SADEGHI M H. Influence of imperfect end boundary condition on the nonlocal dynamics of CNTs [J]. Mechanical Systems and Signal Processing, 2017, 87: 124–135.

[4] CAO G , CHEN X. Buckling of single-walled carbon nanotubes upon bending: Molecular dynamics simulations and finite element method [J]. Physical Review B, 2006, 73(15): 155435.

[5] SRIVASTAVA D, BARNARD S T. Molecular dynamics simulation of large-scale carbon nanotubes on a shared-memory architecture [C]// Proceedings of the 1997 ACM/IEEE Conference on Supercomputing. NY, USA: ACM, 1997: 1–10.

[6] TREACY M M J, EBBESEN T W, GIBSON J M. Exceptionally high Young’s modulus observed for individual carbon nanotubes [J]. Nature, 1996, 381(6584): 678–680.

[7] FATHI R, LOTFAN S. The effect of calibrated nonlocal constant on the modal parameters and stability of axially compressed CNTs [J]. Physica E: Low-dimensional Systems and Nanostructures, 2016, 79: 139–146.

[8] MURMU T, ADHIKARI S. Nonlocal effects in the longitudinal vibration of double-nanorod systems [J]. Physica E: Low-dimensional Systems and Nanostructures, 2010, 43(1): 415–422.

[9] ADHIKARI S, MURMU T, MCCARTHY M. Frequency domain analysis of nonlocal rods embedded in an elastic medium [J]. Physica E: Low-dimensional Systems and Nanostructures, 2014, 59: 33–40.

[10] LOTFAN S, FATHI R, ETTEFAGH M M. Size-dependent nonlinear vibration analysis of carbon nanotubes conveying multiphase flow [J]. International Journal of Mechanical Sciences, 2016, 115: 723–735.

[11] WANG L. A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid [J]. Physica E: Low-dimensional Systems and Nanostructures, 2011, 44(1): 25–28.

[12] MIRRAMEZANI M, MIRDAMADI H R. Effects of nonlocal elasticity and Knudsen number on fluid–structure interaction in carbon nanotube conveying fluid [J]. Physica E: Low-dimensional Systems and Nanostructures, 2012, 44(10): 2005–2015.

[13] ANSARI R, ROUHI H, SAHMANI S. Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics [J]. International Journal of Mechanical Sciences, 2011, 53(9): 786–792.

[14] REZAEE M, LOTFAN S. Non-linear nonlocal vibration and stability analysis of axially moving nanoscale beams with time-dependent velocity [J]. International Journal of Mechanical Sciences, 2015, 96: 36–46.

[15] AYDOGDU M. Axial vibration of the nanorods with the nonlocal continuum rod model [J]. Physica E: Low-dimensional Systems and Nanostructures, 2009, 41(5): 861–864.

[16] ANSARI R, OSKOUIE M F, SADEGHI F, BAZDID- VAHDATI M. Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticity theory [J]. Physica E: Low-dimensional Systems and Nanostructures, 2015, 74: 318–327.

[17] PRADHAN S, MANDAL U. Finite element analysis of CNTs based on nonlocal elasticity and Timoshenko beam theory including thermal effect [J]. Physica E: Low-dimensional Systems and Nanostructures, 2013, 53: 223–232.

[18] HOSEINZADEH M, KHADEM S. A nonlocal shell theory model for evaluation of thermoelastic damping in the vibration of a double-walled carbon nanotube [J]. Physica E: Low-dimensional Systems and Nanostructures, 2014, 57: 6–11.

[19] MURMU T, PRADHAN S. Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM [J]. Physica E: Low-dimensional Systems and Nanostructures, 2009, 41(7): 1232–1239.

[20] MASE G T, MASE G E. Continuum mechanics for engineers [M]. CRC Press, 2010: 40–60.

[21] ERINGEN A C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves [J]. Journal of Applied Physics, 1983, 54(9): 4703–4710.

[22] WANG Q, ARASH B. A review on applications of carbon nanotubes and graphenes as nano-resonator sensors [J]. Computational Materials Science, 2014, 82: 350–360.

[23] WANG Q, WANG C. The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes [J]. Nanotechnology, 2007, 18(7): 075702.

[24] LEE S I, CHUNG J. New non-linear modelling for vibration analysis of a straight pipe conveying fluid [J]. Journal of Sound and Vibration, 2002, 254(2): 313–325.

[25] LJUNG L. System identification: theory for the user [M]. Englewood Cliffs, 1987: 112–140.

[26] LOTFAN S, SALEHPOUR N, ADIBAN H, MASHROUTECHI A. Bearing fault detection using fuzzy C-means and hybrid C-means-subtractive algorithms [C]// International Conference on Fuzzy Systems (FUZZ- IEEE). Istanbul, Turkey: IEEE, 2015: 1–7.

[27] PEETERS B, de ROECK G. Stochastic system identification for operational modal analysis: A review [J]. Journal of Dynamic Systems, Measurement, and Control, 2001, 123(4): 659–667.

[28] ANDERSEN P. Identification of civil engineering structures using vector ARMA models [D]. Aalborg University, 1997.

[29] HAGAN M T, DEMUTH H B, BEALE M H. Neural network design [M]. Boston, 1996: 70–100.

[30] HORNIK K. Approximation capabilities of multilayer feedforward networks [J]. Neural Networks, 1991, 4(2): 251–257.

[31] YETILMEZSOY K, DEMIREL S. Artificial neural network (ANN) approach for modeling of Pb (II) adsorption from aqueous solution by Antep pistachio (Pistacia Vera L.) shells [J]. Journal of Hazardous Materials, 2008, 153(3): 1288–1300.

[32] CAY Y, CICEK A, KARA F, S. Prediction of engine performance for an alternative fuel using artificial neural network [J]. Applied Thermal Engineering, 2012, 37: 217–225.

[33] ROY S, BANERJEE R, BOSE P K. Performance and exhaust emissions prediction of a CRDI assisted single cylinder diesel engine coupled with EGR using artificial neural network [J]. Applied Energy, 2014, 119: 330–340.

[34] DMUTH H, BEALE M. Neural network toolbox for use with Matlab, User’s Guide [M]. Natick, MA. 2000.

[35] LI G, SHI J. On comparing three artificial neural networks for wind speed forecasting [J]. Applied Energy, 2010, 87(7): 2313–2320.

[36] LOTFAN S, GHIASI R A, FALLAH M, SADEGHI M. ANN-based modeling and reducing dual-fuel engine’s challenging emissions by multi-objective evolutionary algorithm NSGA-II [J]. Applied Energy, 2016, 175: 91–99.

[37] SADEGHI M H, LOTFAN S. Identification of non-linear parameter of a cantilever beam model with boundary condition non-linearity in the presence of noise: An NSI- and ANN-based approach [J]. Acta Mechanica, 2017, 228(12): 4451–4469.

(Edited by YANG Hua)

中文导读

基于ARMA和ANN的碳纳米管参数化建模及非局域常数估计

摘要：非局域连续介质力学是研究碳纳米管(CNTs)动态行为的一种新兴理论。在基于该理论的碳纳米管振动行为数学建模中，非局域常数的估计是一个关键步骤。因此，本文提出一种基于振动的非局部参数估计技术，该技术具有较低的仪器和数据分析成本，具有很强的竞争力。为此，利用线性理论和非线性理论建立了碳纳米管的非局域模型。然后，通过数值求解控制方程，得到了CNT在脉冲力作用下的时间响应。利用这些时间响应，采用自回归滑动平均(ARMA)方法建立了CNT的参数模型。引入特征约简技术选择合适的ARMA参数作为识别非局部常数值的特征。在此基础上，通过训练多层感知器(MLP)网络，建立ARMA参数与非局部常数之间的关系。在训练MLP后，基于假设的线性模型和非线性模型，对该方法的性能进行评估。结果表明，在非线性存在或不存在的情况下，该方法都能够以较高的精度估计出非局部参数。

关键词：非局部理论；非局部参数估计；自回归滑动平均；人工神经网络；特征约简

Received date: 2016-12-13; Accepted date: 2017-12-20

Corresponding author: Reza Fathi, PhD Candidate; E-mail: r_fathi@tabrizu.ac.ir; ORCID: 0000-0001-6067-9164