12.1 Introduction

Dynamic analysis of nanostructures is very crucial for engineering design of several electromechanical devices such as nanoprobes, nanooscillators, nanosensors, etc. This analysis is very fundamental as the experimental study in nanoscale is very tedious and expensive. Also, classical mechanics fails to address the nanoscale effect. In this regard, several nonclassical theories have been introduced by researchers to address the small‐scale effect. These theories include strain gradient theory [1], couple stress theory [2], modified couple stress theory [3], micropolar theory, and nonlocal elasticity theory [4]. Investigations related to the dynamical analysis of beams, membranes, nanobeams, nanotubes, nanoribbons, etc., are reported in the literature [5–14].

Akgöz and Civalek [15] analytically studied the static behavior of the Euler–Bernoulli nanobeam under the framework of the modified strain gradient theory and modified couple stress theory. They also investigated the influence of size effect and material parameters on the static response of the beam. Akgöz and Civalek [16] again developed a size‐dependent higher order shear deformation beam using modified strain gradient theory, which can address both the microstructural and shear deformation effects. The dynamical behavior is studied analytically using Navier solution. The static and dynamic analyses of size‐dependent functionally graded microbeams under the framework of Timoshenko beam theory and strain gradient theory are reported in Ansari et al. [17,18]. Kahrobaiyan et al. [19] studied the static and dynamic behaviors of size‐dependent functionally graded simply supported Euler–Bernoulli microbeams using strain gradient theory. Generalized differential quadrature method has been employed by Khaniki and Hosseini‐Hashemi [20] to analyze the buckling behavior of tapered nanobeams using nonlocal strain gradient theory for simply supported boundary condition. Free vibration of a size‐dependent functionally graded Timoshenko beam is studied analytically by Li et al. [21] using strain gradient model implementing Navier's solution. Li et al. [22] again investigated the longitudinal vibration of the nonlocal strain gradient rod by using an analytical method as well as the finite element method. Wave propagation for Euler–Bernoulli beams has been presented by Lim et al. [23] using higher order nonlocal theory as well as strain gradient theory. Some other studies of strain gradient model are also reported in the literature [24–27].

As per the title of the chapter, the dynamical behavior of, in particular, free vibration of Euler–Bernoulli strain gradient nanobeam is investigated using the differential transform method (DTM) for SS and CC boundary conditions. Validation and convergence study of the frequency parameters of strain gradient nanobeam are also conducted. Further, the effects of small‐scale parameters and length‐scale parameters on frequency parameters are reported for the first four modes of frequency parameters through graphical and tabular results.

12.2 Mathematical Formulation of the Proposed Model

The equation of motion of Euler–Bernoulli nanobeam can be expressed as [20]

(12.1)

where M is the bending moment that is defined as , P is the applied compressive force due to mechanical loading, ρ is the mass density, and A is the area of cross section of the beam. For an isotropic beam, the first‐order strain gradient model can be written as [20,23,24]

(12.2)

where σ_xx is the normal stress, E is Young's modulus, ε_xx is the classical strain, e₀a is the nonlocal parameter, and l is the length‐scale parameter. Now, multiplying Eq. (12.2) by z dA and integrating over the area, we may get

(12.3)

Plugging Eq. (12.1) in Eq. (12.3), the nonlocal bending moment may be written as

(12.4)

Inserting Eq. (12.4) in the equation of motion of beam, i.e. Eq. (12.1), the governing equation can be obtained as

(12.5)

Now assuming the equation of motion as sinusoidal, viz., w(x, t) = w₀(x)e^iωt, Eq. (12.5) can be rewritten as

(12.6)

Let us consider the following dimensionless parameters as

Using the above dimensionless parameters in Eq. (12.6), the nondimensional form of governing equation can be expressed as

(12.7)

Now by letting in Eq. (12.7), the nondimensional form of transverse vibration equation of nonlocal strain gradient beam can be reduced to

(12.8)

12.3 Review of the Differential Transform Method (DTM)

DTM is a semianalytical technique that was first introduced by Zhou [28] for solving linear and nonlinear initial value problems arising in electrical circuits. Since then, this technique has been used in many problems arising in different fields of science and engineering. Use of DTM in structural dynamics problem can be found in Chen and Ho [29], Ayaz [30], Ayaz [31], Ozdemir and Kaya [32], Ozdemir and Kaya [33], Balkaya et al. [34], Ozgumus and Kaya [35], Zarepour et al. [36], and Nourifar et al. [37].

Let us consider an analytic function W(X) in a domain D and assume that X = X₀ be any point in that domain. Then, the function W(X) can be expressed by a power series having a center located at X₀. The differential transform of the function W(X) can be written as

(12.9)

where W(X) is the original function and is the transformed function. The inverse differential transformation is defined as

(12.10)

Combining Eqs. (12.9) and (12.10), we obtain

(12.11)

The function W(X) can be expressed by a finite series, and therefore, Eq. (12.11) can be written as

(12.12)

Implementation of differential transform on some basic functions and the boundary conditions are depicted in Tables 12.1–12.2.

Original function	Transformed function
W(X) = W1(X) ± W2(X)
W(X) = αW1(X)
W(X) = W1(X) W2(X)
W(X) = eαX

X = 0		X = 1
Original B.C.	Transformed B.C.	Original B.C.	Transformed B.C.
W(0) = 0		W(1) = 0

12.4 Application of DTM on Dynamic Behavior Analysis

In this section, we will formulate and discuss the dynamic behavior of in particular vibration characteristics of strain gradient nanobeam using DTM to find frequency parameters. Implementing DTM and referring to Table 12.1, the nondimensional form of transverse vibration equation of nonlocal strain gradient beam, i.e. Eq. (12.8), is now reduced into

(12.13)

Now rearranging Eq. (12.13), we obtain the recurrence relation as

(12.14)

Referring to Table 12.2, the simply supported‐simply supported (SS) boundary condition is given as

At X = 0:

(12.15)

At X = 1:

(12.16)

(12.17)

(12.18)

Substituting the values of from Eq. (12.15), into therecurrence relation Eq. (12.14), we obtain

where C₁, C₂, and C₃ are constants. Substituting all the above values of in Eqs. (12.16), (12.17), and (12.18), we will get the system of equations in matrix form as

(12.19)

where , i, j = 1, 2, 3. are the polynomials of λ corresponding to the number of terms as N.

As C₁ ≠ 0, C₂ ≠ 0, and C₃ ≠ 0, Eq. (12.19) implies

(12.20)

Now by solving Eq. (12.20), we get , where i = 1, 2, 3, …N and is the ith mode frequency parameter corresponding to the term N. The value of N for the convergence of the frequency parameter can be obtained from the following relation

(12.21)

where is the ith mode frequency corresponding to N, is the ith mode frequency corresponding to N − 1, and ε is the degree of precision.

Similarly, by referring to Table 12.2, the Clamped‐Clamped (CC) boundary condition can be demonstrated as

At X = 0:

(12.22)

At X = 1:

(12.23)

(12.24)

(12.25)

Referring to the same procedures as that of SS case, we also obtain the frequency parameter for Clamped‐Clamped (CC) boundary condition.

12.5 Numerical Results and Discussion

All the computations for tabular as well as graphical results are carried out using MATLAB tailored code developed by the authors.

12.5.1 Validation and Convergence

Validation and convergence of the present results obtained by DTM are studied in this subsection through graphical as well as tabular results. Setting the length‐scale parameter “l” to zero, the strain gradient model is reduced to Eringen's nonlocal model, and the transverse vibration equation of strain gradient nanobeam, i.e. Eq. (12.8), will be converted into vibration equation for nanobeam. In this regard, β₂ has been assigned zero, and the present results have been compared with other well‐known results reported in the published literature [38]. Tables 12.3 and 12.4 demonstrate the validation of present results with Wang et al. [38] for SS and CC boundary conditions, respectively. In this comparison, the first three frequency parameters are considered for validation, keeping all other parameters the same as Wang et al. [38]. From these Tables 12.3 and 12.4, we may certainly evident that the present results show admirable agreement with Wang et al. [38]. The convergence of the model is also explored through the graphical results, which are illustrated in Figures 12.1 and 12.2. Both Figures 12.1 and 12.2 represent pointwise convergence of SS and CC boundary conditions, which are plotted by taking β₁ = 1 and β₂ = 0.5. From the Figures 12.1 and 12.2, it is revealed that lower mode frequency requires less number of terms than that of higher mode. Approximately 35 number of terms is required for the convergence of third mode frequency of SS boundary condition, whereas 45 number of terms is required for the third mode of CC boundary condition. These figures clearly depict the convergence of the present model by DTM.

	Present			Ref. [38]
0	3.1416	6.2832	9.4248	3.1416	6.2832	9.4248
0.1	3.0685	5.7817	8.0400	3.0685	5.7817	8.0400
0.3	2.6800	4.3013	5.4422	2.6800	4.3013	5.4422
0.5	2.3022	3.4604	4.2941	2.3022	3.4604	4.2941
0.7	2.0212	2.9585	3.6485	2.0212	2.9585	3.6485

	Present			Ref. [38]
0	4.7300	7.8532	10.9956	4.7300	7.8532	10.9956
0.1	4.5945	7.1402	9.2583	4.5945	7.1402	9.2583
0.3	3.9184	5.1963	6.2317	3.9184	5.1963	6.2317
0.5	3.3153	4.1561	4.9328	3.3153	4.1561	4.9328
0.7	2.8893	3.5462	4.1996	2.8893	3.5462	4.1996

12.5.2 Effect of the Small‐Scale Parameter

The effect of small‐scale parameter on the frequency parameter has been investigated through this subsection. The values of small‐scale parameter are considered as 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, and 5. In this regard, both the tabular and graphical results are depicted for the first four frequency parameters of SS and CC boundary conditions. Tables 12.5 and 12.6 represent the tabular results for first four frequency parameters for different small‐scale parameters. Likewise, Figures 12.3 and 12.4 demonstrate the graphical results for SS and CC boundary conditions. It is witnessed that frequency parameters decrease with the increase of small‐scale parameters for both the boundary conditions of all modes. This reduction is very high in case of higher modes than lower modes. Further, to elucidate the nonlocal effect, the response of on frequency ratio, which is defined as the ratio of the frequency parameter calculated using nonlocal theory and local theory, is reported as the graphical report in Figures 12.5 and 12.6. These frequency ratios are less than unity and act as an index to predict small‐scale effect on vibration. All the graphical and tabular results are calculated by taking β₂ = 0.5 with N = 40 for SS case and N = 45 for CC case.

0	4.2869	11.4086	20.6858	31.6968
0.5	3.1415	6.2831	9.4247	12.5663
1.0	2.3610	4.5230	6.7192	8.9274
1.5	1.9532	3.7058	5.4947	7.2955
2.0	1.6995	3.2132	4.7612	6.3201
2.5	1.5235	2.8756	4.2596	5.6536
3.0	1.3925	2.6258	3.8890	5.1614
3.5	1.2901	2.4315	3.6008	4.7788
4.0	1.2074	2.2747	3.3684	4.4703
4.5	1.1387	2.1448	3.1759	4.2147
5.0	1.0805	2.0349	3.0130	3.9985

0	8.6506	17.3399	27.8204	39.8553
0.5	6.0226	9.1041	12.4392	15.5264
1.0	4.4687	6.5319	8.9008	11.0353
1.5	3.6854	5.3482	7.2856	9.0190
2.0	3.2032	4.6362	6.3151	7.8134
2.5	2.8698	4.1486	5.6508	6.9896
3.0	2.6222	3.7881	5.1596	6.3812
3.5	2.4290	3.5076	4.7775	5.9081
4.0	2.2729	3.2814	4.4694	5.5267
4.5	2.1435	3.0939	4.2140	5.2108
5.0	2.0338	2.9353	3.9980	4.9434

12.5.3 Effect of Length‐Scale Parameter

Length‐scale parameter plays very vital role to study vibration characteristics of strain gradient nanobeam. Tables 12.7 and 12.8 show the variation of first four frequency parameters with length‐scale parameter for SS and CC boundary conditions. Similarly, Figures 12.7 and 12.8 represent graphical results for the response of length‐scale parameters on frequency parameters. All these graphical and tabular results are computed by considering β₁ = 0.5. Also, the length‐scale parameters are taken as 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, and 2. From these numerical results, we may clearly note that frequency parameters increase with the increase in length‐scale parameter, and this rise is higher in case of higher modes.

0	2.3022	3.4604	4.2940	4.9820
0.2	2.5019	4.3852	6.2725	8.1937
0.4	2.9175	5.6911	8.4804	11.2785
0.6	3.3629	6.8339	10.2901	13.7397
0.8	3.7863	7.8338	11.8424	15.8351
1.0	4.1802	8.7283	13.2196	17.6886
1.2	4.5467	9.5433	14.4690	19.3676
1.4	4.8894	10.2961	15.6203	20.9133
1.6	5.2119	10.9988	16.6932	22.3530
1.8	5.5169	11.6599	17.7017	23.7059
2.0	5.8070	12.2861	18.6562	24.9859

0	2.2905	3.3153	4.2908	4.9328
0.2	4.2032	6.0618	8.1110	10.0197
0.4	5.4552	8.1923	11.1645	13.9184
0.6	6.5506	9.9397	13.6007	16.9875
0.8	7.5091	11.4388	15.6749	19.5915
1.0	8.3665	12.7688	17.5096	21.8916
1.2	9.1477	13.9755	19.1715	23.9736
1.4	9.8693	15.0874	20.7016	25.8896
1.6	10.5428	16.1236	22.1268	27.6738
1.8	11.1766	17.0977	23.4660	29.3501
2.0	11.7768	18.0196	24.7330	30.9359

12.6 Conclusion

This chapter deals with the study of free vibration of Euler–Bernoulli nanobeam under the framework of the strain gradient model. DTM is applied for the first time to investigate the dynamic behavior of SS and CC boundary conditions. Frequency parameters of strain gradient nanobeam are validated with previously published results of research article showing robust agreement. Also, the convergence of the present results is explored, showing that CC nanobeam requires more points than that of SS boundary condition. Further, the effects of small‐scale parameters and length‐scale parameters on frequency parameters are reported through graphical and tabular results. The frequency parameters decrease with the increase of small‐scale parameters, whereas the trend of frequency parameters is opposite in the case of length‐scale parameter.

Acknowledgment

The first author is very much thankful to the Defence Research & Development Organization (DRDO), Ministry of Defence, New Delhi, India (Sanction Code: DG/TM/ERIPR/GIA/17‐18/0129/020), and the second author is also thankful to the Department of Science and Technology, Government of India, for providing INSPIRE fellowship (IF170207) to undertake the present research work.

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