Der Artikel befasst sich mit dem Schwingungsverhalten einwandiger Kohlenstoffnanoröhren (SWCNTs), die aufgrund ihrer außergewöhnlichen mechanischen, thermischen und elektrischen Eigenschaften entscheidende Komponenten in fortschrittlichen technologischen Anwendungen sind. Das Verständnis der Schwingungseigenschaften von SWCNTs ist von entscheidender Bedeutung, um Stabilität und Leistung in Bereichen wie Luft- und Raumfahrt, Biomedizin und Energiesystemen zu gewährleisten. Die Studie beruht auf der Theorie der nichtlokalen Elastizität, die Größeneffekte im Nanobereich berücksichtigt, indem sie die Spannung an jedem Punkt einer Struktur als von der Dehnungsverteilung innerhalb der Struktur abhängig betrachtet. Dieser Ansatz ist von entscheidender Bedeutung, weil klassische Elastizitätstheorien diese Dynamik im Nanobereich oft nicht erfassen. Der Artikel stellt eine neue Methode vor, die die Initial Value Method (IVM) und die Approximate Transfer Matrix (ATM) kombiniert, um das freie Schwingungsverhalten von SWCNTs auf elastischem Fundament zu analysieren. Diese Methode liefert wertvolle Erkenntnisse darüber, wie Stiftungssteifigkeit und kleinräumige Parameter die Eigenfrequenzen von SWCNTs beeinflussen. Die Forschungsergebnisse bestätigen die rechnerische Effizienz und Robustheit der Methode und bestätigen ihre Fähigkeit, zuverlässige Schwingungsvorhersagen zu treffen. Darüber hinaus betont die Studie, wie wichtig es ist, sowohl die Steifigkeit des Fundaments als auch nichtlokale Effekte bei der genauen Analyse des dynamischen Verhaltens von SWCNTs zu berücksichtigen, was für die Optimierung des Designs von SWCNT-basierten Systemen und die Sicherstellung ihrer dynamischen Stabilität für praktische Anwendungen von entscheidender Bedeutung ist.
KI-Generiert
Diese Zusammenfassung des Fachinhalts wurde mit Hilfe von KI generiert.
Abstract
Single-walled carbon nanotubes (SWCNTs) possess exceptional mechanical and dynamic properties, making them indispensable for diverse applications such as nanoelectromechanical systems, sensor technologies, energy storage devices and composite material design. However, a more accurate, efficient and systematic analysis of their vibrational behaviour under conditions involving elastic foundation stiffness and nanoscale effects remains essential for tailoring their applicability to specific purposes. This study introduces a novel methodological approach to analyse the vibrational behaviour of SWCNTs resting on an elastic foundation, leveraging nonlocal elasticity theory to incorporate small-scale effects. The proposed framework combines the initial value method with the approximate transfer matrix technique to efficiently compute natural frequencies. Vibrational analyses were performed under various boundary conditions, including simply supported, clamped–clamped and clamped–simply supported. Parametric studies revealed that increases in the nonlocal parameter significantly reduce natural frequencies, whereas a higher elastic foundation stiffness enhances the system rigidity and increases vibrational frequencies. This framework offers substantial computational efficiency and accuracy, presenting a robust tool for modelling SWCNT dynamics. The results provide valuable insights for designing effective SWCNT-based structures, contributing to their applications in nano-engineering and other engineering domains requiring precise vibrational analysis.
Hinweise
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1 Introduction
Single-walled carbon nanotubes (SWCNTs) have emerged as critical components with numerous advanced technological applications due to their unique mechanical, thermal and electrical properties. Their high strength-to-weight ratio, combined with their exceptional flexibility and conductivity, makes them ideal for nanoelectromechanical systems (NEMS), sensors, actuators and composite materials. Understanding the vibrational behaviour of SWCNTs is essential for ensuring stability and performance in critical fields such as aerospace, biomedical engineering and energy systems, where precision and reliability are of paramount importance.
Comprehending the vibrational responses of micro/nanostructures under real-world conditions is imperative, as these dynamic characteristics are intrinsically linked to the mechanical stability and operational performance of such systems. However, classical elasticity theories often fail to account for nanoscale size effects. To address these limitations, many size-dependent constitutive theories have been developed, such as nonlocal elasticity theory [1, 2], couple stress theory and modified couple stress theory [3‐5], strain gradient theory and nonlocal strain gradient theory [6, 7], and doublet mechanics [8, 9]. These theories have been extensively applied to analyse the dynamic behaviours of various nanoscale structures, including nanobeams, nanoshells, nanoplates, functionally graded (FG) nanostructures and curved, doubly curved, porous and layered nanostructures [10‐15].
Anzeige
The nonlocal elasticity theory is one of the most prominent size-dependent continuum theories and has attracted significant attention in recent years [16]. It accounts for nanoscale size effects by considering that stress at any point in a structure depends on the strain distribution throughout the entirety of the structure, a feature overlooked by classical elasticity. Early applications of nonlocal elasticity to axial deformation problems in nanostructures were introduced by Tepe and Artan [17]. Many researchers have also investigated the static and dynamic behaviour of Eringen’s nonlocal elasticity models in relation to the bending [18‐24], buckling [25‐28], vibration [29‐32] and wave propagation [33‐35] of micro- and nanostructures.
In an early contribution, Reddy reformulated classical beam theories, including Euler–Bernoulli, Timoshenko and higher-order theories, to study the influence of nonlocal effects on bending, buckling and vibration responses in beams [36]. Wang et al. applied nonlocal Timoshenko beam theory to analyse high-frequency vibrations in micro- and nanobeams, revealing the significance of transverse shear deformation and rotary inertia [37]. Building on this, Reddy and Pang utilised the nonlocal Euler and Timoshenko beam models to analyse the dynamic behaviours of SWCNTs [38]. Murmu and Pradhan further extended this theory to investigate the thermal vibrations of SWCNTs embedded in elastic media, providing insights into their temperature-dependent dynamic behaviour [39]. Aydogdu expanded on these foundations by developing analytical models for bending, buckling and axial vibrations in nanostructures, demonstrating that nonlocal parameters exerted a significant influence on the stability and frequency characteristics of these systems [40, 41]. Yang et al. explored the nonlinear vibration behaviour of SWCNTs under various boundary conditions [42] and Arghavan and Singh analysed the free and forced vibrations of SWCNTs using space frame elements with extensional, bending and torsional stiffness properties, employing advanced computational techniques, such as the Newmark integration method and fast Fourier transform (FFT), to capture transient response characteristics [43]. Ebrahimi et al. applied nonlocal Timoshenko beam theory to investigate the vibrational characteristics of nanobeams, emphasising the impact of small-scale effects and material properties on their dynamic responses [44]. Boumia et al. studied the influence of chirality on natural frequencies in SWCNT [45]. Wu and Lai applied the mixed variational principle to free vibrations of SWCNT [46]. Su and Cho conducted a comprehensive analysis of the vibrational behaviour of SWCNTs embedded in an elastic medium using the nonlocal Timoshenko beam model, examining the natural frequencies and mode shapes of SWCNTs under various boundary conditions and embedded elastic medium stiffness, with a particular focus on the impact of nonlocal effects on vibration modes [47]. Noureddine et al. examined the vibrational properties of SWCNTs using the nonlocal Euler–Bernoulli beam model, focussing on how chiral angle and chiral index influence natural frequencies. They found that chirality significantly impacts frequency characteristics, providing guidance for designing chiral SWCNT-based nanodevices [48].
To solve complex vibrational problems, researchers have integrated nonlocal elasticity theory with numerical and analytical methods. The Finite Element Method (FEM) approach is particularly suitable for flexible modelling under complex boundary conditions. Eltaher et al. employed FEM in combination with nonlocal elasticity to investigate the effects of boundary conditions and slenderness ratios on natural frequencies in nanobeams, providing critical insights into their dynamic behaviour [49]. Similarly, Attia et al. applied FEM to accurately predict the frequencies of composite shells, emphasising its importance for aerospace applications [50]. Mojahedi et al. expanded FEM applications by integrating it with the Galerkin method within a nonlocal strain gradient framework. Their study revealed how nonlocal parameters influence deflection and frequency in nonlinear vibration analyses [51]. Kharche et al. examined the vibration analysis of SWCNTs for sensor applications, using FEM to study frequency and frequency shifts. They demonstrated the effects of boundary conditions, nanotube structure (zigzag versus armchair) and thickness on fundamental frequencies, showing that bridged SWCNTs exhibited higher sensitivity, which is essential for the development of mass-sensitive nanosensors [52]. A recent approach utilising a post-processing framework to compute nonlocal stresses from classical finite element solutions has been demonstrated for multi-phase nanocomposites [53]. Differential Transform Method (DTM) is another valuable tool for vibrational analysis. Ni et al. applied DTM to fluid-conveying pipes, showing that foundation elasticity affects natural frequencies and critical velocities [54]. Ebrahimi et al. extended DTM to the thermomechanical vibration analysis of functionally graded nanobeams, revealing significant effects of temperature and material gradient [55]. The Differential Quadrature Method (DQM) is a high-accuracy numerical approach widely employed in nanoscale applications, particularly for solving complex vibrational problems. For example, Belhadj et al. utilised DQM in conjunction with nonlocal elasticity theory to study the vibrations of SWCNTs, highlighting the critical role of nonlocal parameters in accurately capturing vibrational modes [56], and Balkaya and Kaya applied DQM to analyse fluid-conveying pipes, showing that foundation stiffness significantly influences critical velocities [57].
In addition to FEM, DTM and DQM, researchers have employed the Initial Value Method (IVM) and the Approximate Transfer Matrix (ATM) approach to address a wide range of problems in nanoscale mechanics, including the buckling and vibrational behaviours of nanostructures such as single-walled and double-walled CNTs, curved geometries and nanobars. Several applications of IVM and ATM in this context can be found in the literature [58‐61]. Additionally, Tufekci and Yigit [62] and Tufekci et al. [63] examined the in-plane vibrations of circular arches and spatial frame structures, respectively, utilising initial value methods to analyse how axial extension, transverse shear and rotational inertia affect vibrational modes. These methods provide further understanding of the complex dynamic responses of macro- and nano-sized structures [64]. Jena et al. employed numerical approaches, specifically the Chebyshev–Ritz and Navier methods, to analyse the vibrational characteristics of an Euler–Bernoulli nanobeam resting on a Winkler–Pasternak elastic foundation. Utilising nonlocal elasticity theory, they examined the effects of magnetic and hygroscopic environments on frequency parameters, showing that increased nonlocal parameters reduce the frequency values, while higher foundation stiffness coefficients, such as the Winkler modulus, lead to an increase in frequency [65]. Artan et al. analysed the vibration of isotropic shear beams (ISBs) using unified shear deformation theory and the initial value method. Governing equations were derived using the virtual work principle, and the first five natural frequencies under various end conditions were calculated with high precision. The use of approximate transfer matrices further streamlined the analysis of shear stress distributions [66].
Anzeige
Several studies have reported paradoxes and mathematical inconsistencies in size-dependent elasticity theories, particularly in Eringen’s nonlocal elasticity model. Notably, when analysing cantilever nanobeams subjected to concentrated tip loads, the nonlocal solution reduces to the local solution, contradicting the expected size-dependent behaviour and revealing an inappropriate formulation [67‐69]. To address these issues, several mathematically consistent and physically meaningful models have been proposed, such as the stress- and reaction-driven nonlocal integral models [70‐73]. These advanced models have been widely applied to various nanoscale structural problems, including nanobeams resting on nonlocal foundations, structures under high-frequency vibrations and systems incorporating surface energy effects. For instance, a dedicated finite element formulation for stress-driven beams was introduced by Pinnola et al. to enhance computational accuracy in capturing nonlocal behaviours [74]. Apuzzo et al. proposed a closed-form solution for higher-mode vibration analysis [75], and Penna et al. incorporated surface energy effects into buckling analysis of functionally graded nanobeams [76].
Although Eringen’s nonlocal model is known to have certain limitations under specific boundary conditions, it remains widely used in the literature due to its historical significance and its ability to capture scale-dependent behaviour when applied with appropriate boundary and initial conditions. In the present study, the boundary and initial conditions were carefully defined, and the modelling approach was structured in accordance with both kinematic and equilibrium conditions. However, it is known that Eringen’s model may lead to paradoxical results under cantilever boundary conditions, where an increase in the nonlocal parameter may result in physically inconsistent rises in natural frequencies. To avoid such inconsistencies, this particular boundary condition was excluded from the scope of the present study. Therefore, the results presented herein should be interpreted within the theoretical framework and known limitations of Eringen’s model. Moreover, the initial value method combined with the approximate transfer matrix (matricant) approach employed in this study solves the governing equations sequentially and systematically, thereby enhancing the reliability of the numerical results.
Furthermore, since the present study also involves nanobeams supported by an elastic foundation, it is important to highlight the modelling assumptions and limitations of the foundation model adopted. The Winkler elastic foundation model has been shown to be inadequate for accurately capturing size-dependent behaviour at the nanoscale, it is primarily due to its fundamental limitations in representing nonlocal interactions. In response, several advanced nonlocal foundation models have been proposed in recent literature. For example, Barretta et al. [77] proposed a modelling framework in which a stress-driven law governs the nanobeam and a displacement-driven nonlocal model represents the foundation, enhancing the representation of beam–foundation interactions. Additionally, Pinnola et al. [78] developed a reaction-driven nonlocal foundation model by combining the Winkler and Wieghardt elastic laws through a convex formulation, thereby capturing size-dependent behaviours more accurately. In this study, the elastic foundation is modelled using the Winkler approach, which provides a straightforward yet robust means of representing the local support of the beam. Mechanically, the Winkler model assumes that the reaction force at any point on the nanobeam is directly proportional to its local deflection, idealising the supporting medium as an array of independent, linear springs. As a result, it enables a systematic investigation of interactions between nonlocality and elastic support, explaining how these phenomena govern the vibrational behaviour of nanobeams, and thereby enhancing the predictive capability of the model for nano-engineered applications.
Building upon this modelling framework, the present work aims to investigate the free vibration behaviour of single-walled carbon nanotubes (SWCNTs) resting on elastic foundations. By incorporating the effects of foundation stiffness and small-scale parameters, the proposed model provides valuable insights into the influence of these factors on the natural frequencies of SWCNTs. This methodology is particularly important for optimising the design of SWCNT-based systems and ensuring their dynamic stability for practical applications, where the precise prediction of vibrational behaviour is essential for maintaining structural integrity and performance.
2 Basic equations
The constitutive equations for time-harmonic vibrations of a linear elastic, isotropic and homogeneous EBB beam on an elastic foundation are as follows:
$$\begin{aligned} \frac{\mathrm{{d}}{T(z,t)}}{\mathrm{{d}}z}= & (k-\rho A \omega ^2)v(z,t), \end{aligned}$$
(4)
where v(z, t) is the transverse deflection of a point z\((0 \le z \le L)\) at time t, L is the beam length, \(\varphi \) is the rotation, M is the bending moment, T is the shear force, \(\rho \) is the mass density (mass/length) of the material of the beam, EI is the bending rigidity, k is the spring constant, A is the area of the cross section and \(\omega \) is the natural frequency of the beam. The geometrical compatibility condition Eq. (1) and the equilibrium conditions Eqs. (3) and (4) are valid in nonlocal elasticity. On the other hand, Eq. (2) takes a different form in nonlocal elasticity, such that the following relationship is valid [79]:
where \(\gamma \) is the nonlocal parameter. Previous studies have reported that the nonlocal integral formulation adopted in Eq. (5) may exhibit ill-posedness issues under certain boundary conditions and loading scenarios [68‐70]; however, such issues do not arise in the present analysis due to the careful definition of boundary and initial conditions. After differentiating the nonlocal form of Eq. (3) and substituting the resulting expression into Eq. (5) and using Eq. (4), the nonlocal form of Eq. (2) becomes as follows:
where \(\textbf{Y}(z,0)\) is termed a principal matrix (or transfer matrix), and \(\textbf{y}_0\) is a column matrix consisting of initial values depending on the boundary conditions applied at the end (\(z=0\)) of the beam. Due to the inherent complexity of initial value problems (IVPs), it is essential to derive the principal matrix; this requires a systematic approach to expressing the matrix. One effective method is Gantmacher’s approximate transfer matrix (ATM), also known as the matricant (Gantmacher 1959), which serves as a powerful tool for analysing systems based on initial conditions specified at different points. The matricant can be obtained using the method of successive approximations [80] as follows:
Here, \(\textbf{Y}(z, z_1)\) is termed the matricant and \(\textbf{A}\) is the matrix defined in Eq. (12). The matricant can be computed by taking several terms from the series. To calculate the value of this matrix at a specific point with a high degree of accuracy, the following well-known property should be utilised:
This feature allows the division of the interval between 0 and z into the desired number of segments, facilitating sequential progress and improving convergence. If the matricant Y(z, 0) is known, the solution of the initial value problem at z can be readily determined. The matricant can be efficiently evaluated using successive approximations, providing a practical and accurate framework for frequency analysis that may offer advantages over other numerical approaches. This strategy ensures systematic and reliable results within the Initial Value Method (IVM), optimising the overall precision and efficiency for analysing nanostructures.
4 Convergence, validation and nonlocal frequency analysis in SWCNT vibrations
In this section, a comprehensive investigation is carried out to evaluate the vibrational behaviour of single-walled carbon nanotubes (SWCNTs) using both the Initial Value Method (IVM) and the Approximate Transfer Matrix (ATM) approach within the framework of nonlocal elasticity theory. The study begins with a convergence analysis to ensure the accurate computation of natural frequencies for different boundary conditions, including simply supported (SS), clamped–clamped (CC) and clamped–simply supported (CS), by initially setting the nonlocal parameter to zero. Subsequently, the proposed methodology is validated by comparing the obtained frequencies in the local elasticity case with benchmark results. A parametric study is then performed to investigate the influence of the nonlocal parameter on the natural frequencies of SWCNTs under various boundary conditions and vibration modes. The nonlocal parameter is defined as the ratio \(\gamma /L\), varying between \(1/20\) and \(1/10\). The geometric and material properties are set as follows: width-to-thickness ratio \(b/h = 2/3\), slenderness ratio \(h/L = 0.03\), Poisson’s ratio \(\nu = 0.3\), Young’s modulus \(E = 2.1 \times 10^{11} \, \text {Pa}\) and density \(\rho = 7860 \, \text {kg/m}^3\) [81]. The non-dimensional elastic foundation stiffness is given by \(k = \dfrac{k_w L^4}{EI}\), while the corresponding non-dimensional natural frequency is defined as \(\Omega _n = \omega _n L^2 \sqrt{\dfrac{\rho A}{EI}}\), where \(n = 1, 2, \dots \). This formulation enables a systematic evaluation of how scale effects, characterised by \(\gamma \), influence the vibrational response of the system.
4.1 Simply supported beam
If the initial values \(v^{nl}(0) = M^{nl}(0)=0\) and the end conditions \(v^{nl}(L) = M^{nl}(L) = 0\) are used in Eq. (14), the system can be written as follows:
Table 1 presents the dimensionless frequency values for six vibration modes obtained using various combinations of term and interval numbers using the matricant approach. The table includes results for 6, 8 and 9 terms across 5, 6 and 8 intervals, along with the exact analytical solutions for comparison. As observed, the configuration using nine terms and eight intervals yields frequency values that exactly match the analytical solution for all six modes, confirming full convergence and high numerical precision. Table 2 shows the convergence behaviour of the first four natural frequencies under simply supported (SS) boundary conditions for increasing numbers of terms, while the number of intervals is fixed at eight. It is observed that increasing the number of terms from nine to ten does not lead to any noticeable change in the frequency values up to four significant digits. Therefore, using nine terms and eight intervals provides an optimal balance between accuracy and computational efficiency.
Table 1
Convergence of the dimensionless frequency values \(\Omega _i\) for varying number of terms and intervals
Mode (\(\Omega _i\))
6 Terms
8 Terms
9 Terms
Exact
5 Intervals
\(\Omega _1\)
9.8696
9.8696
9.8696
9.8696
\(\Omega _2\)
39.4698
39.4755
39.4784
39.4784
\(\Omega _3\)
88.4686
88.7073
88.8268
88.8264
\(\Omega _4\)
154.4760
156.6367
157.9660
157.9140
\(\Omega _5\)
236.9510
244.2213
247.8560
246.7420
\(\Omega _6\)
428.5610
387.2297
366.5690
355.3570
6 Intervals
\(\Omega _1\)
9.8696
9.8696
9.8696
9.8696
\(\Omega _2\)
39.4763
39.4784
39.4784
39.4784
\(\Omega _3\)
88.7258
88.8264
88.8264
88.8264
\(\Omega _4\)
156.6600
157.9190
157.9180
157.9140
\(\Omega _5\)
240.2960
246.7970
246.8760
246.7420
\(\Omega _6\)
341.2090
355.6690
356.9130
355.3570
8 Intervals
\(\Omega _1\)
9.8696
9.8696
9.8696
9.8696
\(\Omega _2\)
39.4763
39.4784
39.4784
39.4784
\(\Omega _3\)
88.7258
88.8266
88.8264
88.8264
\(\Omega _4\)
156.6600
157.9190
157.9140
157.9140
\(\Omega _5\)
240.2960
246.7970
246.7420
246.7420
\(\Omega _6\)
341.2090
355.6690
355.3570
355.3570
Table 2
Convergence of the first four natural frequencies for the SS condition at \(\gamma /L=0\) with varying number of terms. (FRS: Frequencies)
Number of terms
FRS
4
6
8
9
10
12
14
\(\Omega _1\)
9.8697
9.8696
9.8696
9.8696
9.8696
9.8696
9.8696
\(\Omega _2\)
39.4886
39.4783
39.4784
39.4784
39.4784
39.4784
39.4784
\(\Omega _3\)
89.0696
88.8216
88.8265
88.8264
88.8264
88.8264
88.8264
\(\Omega _4\)
160.1477
157.8379
157.9153
157.9136
157.9137
157.9137
157.9137
Table 3 presents a comparison of the first four natural frequencies obtained in the present study using the ATM method with those derived from other approaches, including the DTM, the DQM, exact solutions and the EBT, for \(\gamma /L = 0\) and \(k=0\). The results demonstrate an excellent agreement between the ATM approach and the exact solutions, with frequencies matching up to four significant digits in most cases. This close agreement underscores the accuracy and reliability of the proposed ATM approach, confirming its capability to accurately capture natural frequencies. By providing a systematic and robust framework, the ATM method serves as an alternative computational tool that complements existing techniques while maintaining comparable precision.
Table 3
Comparison of the first four natural frequencies for the SS condition at \(\gamma /L=0\). (FRS: Frequencies)
Nonlocal frequencies for different \(\gamma /L\) values under the SS condition without elastic foundation. (FRS: Frequencies)
FRS/ \(\gamma /L\)
1/20
1/18
1/16
1/14
1/12
1/10
\(\Omega _1\)
9.7501
9.7226
9.6847
9.6301
9.5478
9.4159
\(\Omega _2\)
37.6635
37.2729
36.7466
36.0174
34.9743
33.4277
\(\Omega _3\)
80.3517
78.6921
76.5353
73.6852
69.8566
64.6414
\(\Omega _4\)
133.7110
129.4810
124.1890
117.5170
109.0590
98.3292
\(\Omega _5\)
194.0460
185.9060
176.0710
164.1700
149.7880
132.5060
\(\Omega _6\)
258.5630
245.3790
229.9260
211.8510
190.8080
166.5130
Table 4 presents the first six natural frequencies (\(\Omega _1\) to \(\Omega _6\)) for various values of \(\gamma /L\) under SS condition, showing that as \(\gamma /L\) increases (from 1/20 to 1/10), there is a consistent decrease in natural frequencies across all modes. This trend demonstrates the significant influence of nonlocal effects, where larger \(\gamma /L\) ratios reduce the effective stiffness of the structure, resulting in lower natural frequencies. Additionally, higher-order modes show a more noticeable reduction than lower-order modes, emphasising the increased sensitivity of higher modes to nonlocal effects. These findings underline the importance of considering nonlocal parameters in vibrational analysis, particularly for systems in which \(\gamma /L\) values play a critical role in defining the dynamic response.
4.2 Clamped–simply supported beam
If the initial values \(v^{nl}(0) = \varphi ^{nl}(0) = 0\) and the end conditions \(v^{nl}(L) = M^{nl}(L) = 0\) are used in Eq. (14), the system can be written as follows:
Table 5 illustrates the convergence behaviour of the first four natural frequencies for the clamped–simply supported (CS) boundary condition. The matricant is computed by dividing the solution interval into eight segments using Eq. (16), and the natural frequencies are obtained via Eq. (20). As can be seen from the table, increasing the number of terms from nine to ten results in no significant change in the frequencies up to four significant digits, indicating convergence. Therefore, employing nine terms and eight segments ensures an effective balance between computational efficiency and accuracy.
Table 5
Convergence of the first four natural frequencies for the CS condition at \(\gamma /L=0\) with varying number of terms. (FRS: Frequencies)
Number of terms
FRS
4
6
8
9
10
12
14
\(\Omega _1\)
15.4185
15.4182
15.4182
15.4182
15.4182
15.4182
15.4182
\(\Omega _2\)
49.9907
49.9646
49.9649
49.9649
49.9649
49.9649
49.9649
\(\Omega _3\)
104.6993
104.2372
104.2479
104.2477
104.2477
104.2477
104.2477
\(\Omega _4\)
181.8339
178.1362
178.2734
178.2696
178.2697
178.2697
178.2697
Table 6 compares the first four natural frequencies (\(\Omega _1, \Omega _2, \Omega _3\) and \(\Omega _4\)) from this study with results from other methods, including DTM, DQM, exact solutions and EBT, for \(\gamma /L = 0\) and \(k=0\) under CS conditions. The results obtained show strong agreement up to four decimal places, underscoring the accuracy of the present approach.
Table 6
Comparison of the first four natural frequencies for the CS condition at \(\gamma /L=0\). (FRS: Frequencies)
Table 7 shows the first six natural frequencies for various values of \(\gamma /L\) under CS conditions without an elastic foundation. As \(\gamma /L\) increases, a consistent reduction in frequency values is observed across all modes. This trend indicates that higher \(\gamma /L\) ratios lead to lower natural frequencies, reflecting the influence of nonlocal effects on the structural response.
Table 7
Nonlocal frequencies for different \(\gamma /L\) values under the CS condition without an elastic foundation. (FRS: Frequencies)
FRS/ \(\gamma /L\)
1/20
1/18
1/16
1/14
1/12
1/10
\(\Omega _1\)
15.2009
15.1512
15.0825
14.9840
14.8358
14.5992
\(\Omega _2\)
47.4829
46.9523
46.2394
45.2554
43.8551
41.7947
\(\Omega _3\)
93.8018
91.7753
89.1508
85.6978
81.0863
74.8518
\(\Omega _4\)
150.0100
145.1260
139.0360
131.3900
121.7490
109.5950
\(\Omega _5\)
212.5090
203.4230
192.4800
179.2880
163.4140
144.4270
\(\Omega _6\)
278.6530
264.2630
247.4440
227.8300
205.0710
178.8810
4.3 Clamped–clamped supported beam
If the initial values \(v^{nl}(0) = \varphi ^{nl}(0) = 0\) and the end conditions \(v^{nl}(L) = \varphi ^{nl}(L) = 0\) are used in Eq. (14), the system can be written as follows:
A similar convergence behaviour to that observed in the SS and CS boundary conditions is also evident for the clamped–clamped (CC) configuration, as shown in Table 8. The matricant is computed by dividing the solution interval into eight segments using Eq. (16), and the natural frequencies are obtained via Eq. (22). Consistent with the previous results, increasing the number of terms from nine to ten results in no significant change in the frequency values up to four significant digits, confirming that convergence has been achieved. Therefore, employing nine terms and eight segments ensures both numerical accuracy and computational efficiency in the current analysis.
Table 8
Convergence of the first four natural frequencies for the CC condition at \(\gamma /L=0\) with varying number of terms. (FRS: Frequencies)
Number of terms
FRS
4
6
8
9
10
12
14
\(\Omega _1\)
22.3743
22.3733
22.3733
22.3733
22.3733
22.3733
22.3733
\(\Omega _2\)
61.7317
61.6720
61.6728
61.6728
61.6728
61.6728
61.6728
\(\Omega _3\)
121.7030
120.8820
120.9038
120.9034
120.9034
120.9034
120.9034
\(\Omega _4\)
205.4077
199.6329
199.8657
199.8594
199.8594
199.8594
199.8594
Table 9 presents a comparison of the first four natural frequencies (\(\Omega _1, \Omega _2, \Omega _3\) and \(\Omega _4\)) obtained in this study with those calculated using other methods, including DTM, DQM, exact solutions and the EBT, for \(\gamma /L = 0\) and \(k=0\). The comparison reveals a high level of agreement, with values matching up to four decimal places in most instances. This close agreement confirms the robustness and effectiveness of the present method in capturing natural frequencies.
Table 9
Comparison of the first four natural frequencies for the CC condition at \(\gamma /L=0\). (FRS: Frequencies)
Table 10 presents the first six natural frequencies for various \(\gamma /L\) values under CC boundary conditions without an elastic foundation. It is observed that, as \(\gamma /L\) increases, the frequency values consistently decrease across all modes.
Table 10
Nonlocal frequencies for different \(\gamma /L\) values under the CC condition without an elastic foundation. (FRS: Frequencies)
FRS/ \(\gamma /L\)
1/20
1/18
1/16
1/14
1/12
1/10
\(\Omega _1\)
22.0367
21.9599
21.8537
21.7016
21.4731
21.1090
\(\Omega _2\)
58.3969
57.6998
56.7650
55.4781
53.6537
50.9832
\(\Omega _3\)
108.2480
105.8120
102.6670
98.5462
93.0700
85.7164
\(\Omega _4\)
167.1790
161.5860
154.6350
145.9410
135.0290
121.3480
\(\Omega _5\)
231.7280
221.6380
209.5220
194.9660
177.5180
156.7380
\(\Omega _6\)
299.3990
283.7450
265.4960
244.2760
219.7270
191.5550
5 Results and discussion
5.1 Effects of small scale
Herein, the impact of nonlocal effects on the vibrational behaviour of nanobeams without an elastic foundation is examined to analyse how variations in the nonlocal parameter \(\gamma /L\) influence the first six mode frequencies and overall vibrational characteristics across the SS, CC and CS boundary conditions. As shown in Figure 1, the frequency ratios decrease across all modes as the \(\gamma /L\) ratio increases. This decrease becomes more pronounced at higher modes. Figure 2 illustrates the nonlocal effects on the first mode frequency \(\Omega _1\) for different boundary conditions (SS, CS and CC). The nonlocal to local ratio of the first mode natural frequency \(\Omega _1\) is plotted against the \(\gamma /L\) ratio, clearly indicating that, as the \(\gamma /L\) ratio increases, the nonlocal effects become more pronounced, with the most significant reduction in frequency ratios observed under the CC boundary condition.
Fig. 1
Nonlocal to local ratios of the first six natural frequencies under \(k=0\) for different boundary conditions: (a) SS, (b) CC and (c) CS
Figure 3 presents the variation in mode frequencies of nanobeams with increasing \(\gamma /L\) ratio under the SS, CC and CS boundary conditions. As the mode number increases, a consistent rise in frequency is observed for all boundary conditions; however, a clear decrease in frequency values occurs with an increasing \(\gamma /L\) ratio, which reflects the influence of nonlocal effects. This reduction in frequency is associated with the softening of the nanobeam’s stiffness due to long-range interactions accounted for by nonlocal elasticity theory. The CC boundary condition produces the highest frequencies, as the clamped ends provide greater rigidity, while the SS boundary condition results in the lowest frequencies, corresponding to lower structural constraints. These results highlight the critical role of both nonlocal effects and boundary conditions in governing the vibrational response of nanobeams.
Fig. 3
Effects of \(\gamma /L\) on mode frequencies for the SS, CC and CS boundary conditions with \(k=0\)
The combined effects of the elastic foundation and the nonlocal parameter \(\gamma /L\) on the vibrational behaviour of nanobeams are investigated. The analysis is focussed on how these factors influence the first six mode frequencies across different boundary conditions. The presence of the elastic foundation, characterised by the stiffness parameter \(k\), is expected to significantly alter natural frequencies, particularly when coupled with varying \(\gamma /L\) ratios. Tables 11, 12 and 13 present the detailed results, showing how the interaction between the elastic foundation and nonlocal effects impacts the vibrational characteristics of the system.
Table 11
Frequencies for different \( k \) values and \( \gamma /L \) ratios for SS configuration
\( k \)
\( \gamma /L \)
\( \Omega _1 \)
\( \Omega _2 \)
\( \Omega _3 \)
\( \Omega _4 \)
\( \Omega _5 \)
\( \Omega _6\)
10
0
10.3638
39.6049
88.8827
157.9450
246.7600
355.3150
1/20
10.2500
37.7960
80.4139
133.7480
194.0720
258.5820
1/18
10.2240
37.4068
78.7556
129.5200
185.932
245.4000
1/16
10.1879
36.8824
76.6006
124.2300
176.0990
229.9480
1/14
10.1360
36.1559
73.7530
117.5590
164.2000
211.8740
1/12
10.0579
35.1169
69.9281
109.1040
149.8210
190.8340
1/10
9.9327
33.5769
64.7187
98.3800
132.5440
166.5430
25
0
11.0639
39.7938
88.9671
157.9930
246.7900
355.3360
1/20
10.9573
37.9940
80.5071
133.8040
194.1100
258.6110
1/18
10.9330
37.6067
78.8508
129.5780
185.9730
245.4300
1/16
10.8992
37.0852
76.6985
124.2900
176.1420
229.9810
1/14
10.8508
36.3628
73.8546
117.6230
164.2460
211.9100
1/12
10.7778
35.3299
70.0353
109.1730
149.8710
190.8730
1/10
10.6611
33.7996
64.8345
98.4562
132.6010
166.5880
50
0
12.1412
40.1067
89.1074
158.0720
246.8410
355.3710
1/20
12.0442
38.3215
80.6622
133.8980
194.1750
258.6600
1/18
12.0220
37.9377
79.0091
129.6740
186.0400
245.4810
1/16
11.9914
37.4207
76.8613
124.3910
176.2130
230.0350
1/14
11.9474
36.7049
74.0237
117.7290
164.3220
211.9690
1/12
11.8811
35.6819
70.2135
109.2870
149.9550
190.9390
1/10
11.7753
34.1674
65.0270
98.5831
132.6950
166.6630
100
0
14.0502
40.7252
89.3876
158.2300
246.9420
355.4420
1/20
13.9665
38.9685
80.9715
134.0840
194.3030
258.7560
1/18
13.9474
38.5910
79.3249
129.8670
186.1740
245.5830
1/16
13.9210
38.0829
77.1859
124.5910
176.3550
230.1440
1/14
13.8831
37.3798
74.3606
117.9410
164.4740
212.0870
1/12
13.8261
36.3758
70.5687
109.5160
150.1210
191.0700
1/10
13.7353
34.8914
65.4103
98.8364
132.8830
166.8130
200
0
17.2456
41.9350
89.9452
158.5460
247.1450
355.5820
1/20
17.1774
40.2311
81.5867
134.4570
194.5600
258.9500
1/18
17.1619
39.8656
79.9528
130.2510
186.4430
245.7860
1/16
17.1404
39.3740
77.8309
124.9920
176.6380
230.3610
1/14
17.1096
38.6943
75.0300
118.3640
164.7780
212.3220
1/12
17.0634
37.7253
71.2737
109.9720
150.4540
191.3310
1/10
16.9900
36.2961
66.1703
99.3410
133.2590
167.1120
Table 12
Frequencies for different \( k \) values and \( \gamma /L \) ratios for the CS configuration
\( k \)
\( \gamma /L \)
\( \Omega _1 \)
\( \Omega _2 \)
\( \Omega _3 \)
\( \Omega _4 \)
\( \Omega _5 \)
\( \Omega _6 \)
10
0
15.7392
50.0648
104.2960
178.2980
272.0480
385.5370
1/20
15.5263
47.5881
93.8551
150.0430
212.5330
278.6710
1/18
15.4777
47.0587
91.8297
145.1600
203.4470
264.2820
1/16
15.4105
46.3474
89.2068
139.0720
192.5060
247.4640
1/14
15.3140
45.3657
85.7562
131.4280
179.3160
227.8520
1/12
15.1691
43.9690
81.1479
121.7900
163.4440
205.0950
1/10
14.9378
41.9142
74.9186
109.6400
144.4610
178.9090
25
0
16.2087
50.2144
104.3680
178.3400
272.0760
385.5560
1/20
16.0021
47.7455
93.9349
150.0930
212.5680
278.6980
1/18
15.9549
47.2178
91.9114
145.2120
203.4840
264.3100
1/16
15.8897
46.5089
89.2908
139.1260
192.5450
247.4940
1/14
15.7962
45.5308
85.8436
131.4860
179.3580
227.8850
1/12
15.6557
44.1392
81.2403
121.8520
163.4900
205.1320
1/10
15.4317
42.0927
75.0186
109.7090
144.5130
178.9510
50
0
16.9623
50.4627
104.4870
178.4100
272.1220
385.5890
1/20
16.7651
48.0066
94.0679
150.1770
212.6270
278.7430
1/18
16.7200
47.4818
92.0473
145.2980
203.5460
264.3580
1/16
16.6578
46.7769
89.4307
139.2160
192.6100
247.5450
1/14
16.5686
45.8045
85.9891
131.5810
179.4270
227.9400
1/12
16.4347
44.4215
81.3940
121.9550
163.5660
205.1930
1/10
16.2215
42.3887
75.1850
109.8230
144.6000
179.0210
100
0
18.3772
50.9557
104.7260
178.5500
272.2140
385.6530
1/20
18.1953
48.5245
94.3333
150.3430
212.7440
278.8320
1/18
18.1538
48.0054
92.3185
145.4700
203.6680
264.4520
1/16
18.0965
47.3084
89.7099
139.3950
192.7400
247.6460
1/14
18.0144
46.3471
86.2793
131.7700
179.5670
228.0490
1/12
17.8914
44.9808
81.7006
122.1590
163.7190
205.3150
1/10
17.6957
42.9744
75.5168
110.0500
144.7720
179.1600
200
0
20.9218
51.9277
105.2030
178.8300
272.3970
385.7830
1/20
20.7622
49.5442
94.8619
150.6750
212.9790
279.0120
1/18
20.7258
49.0359
92.8585
145.8130
203.9140
264.6410
1/16
20.6757
48.3537
90.2655
139.7540
192.9990
247.8480
1/14
20.6039
47.4136
86.8569
132.1490
179.8450
228.2680
1/12
20.4964
46.0790
82.3103
122.5680
164.0240
205.5580
1/10
20.3258
44.1225
76.1760
110.5030
145.1170
179.4390
Table 13
Frequencies for different \( k \) values and \( \gamma /L \) ratios for the CC configuration
\( k \)
\( \gamma /L \)
\( \Omega _1 \)
\( \Omega _2 \)
\( \Omega _3 \)
\( \Omega _4 \)
\( \Omega _5 \)
\( \Omega _6\)
10
0
22.5957
61.7538
120.945
199.8840
298.5710
416.9910
1/20
22.2625
58.4824
108.294
167.2090
231.7500
299.4160
1/18
22.1864
57.7863
105.859
161.6170
221.6600
283.7630
1/16
22.0814
56.8530
102.716
154.6680
209.5460
265.5150
1/14
21.9308
55.5682
98.5969
145.9750
194.9910
244.2960
1/12
21.7047
53.7468
93.1237
135.0660
177.5460
219.7270
1/10
21.3446
51.0811
85.7747
121.3890
156.7000
191.5810
25
0
22.9252
61.8752
121.0070
199.9220
298.5960
417.0090
1/20
22.5968
58.6105
108.3630
167.2540
231.7820
299.4410
1/18
22.5219
57.9160
105.9300
161.6640
221.6940
283.7890
1/16
22.4184
56.9848
102.7890
154.7160
209.5820
265.5430
1/14
22.2701
55.7030
98.6729
146.0270
195.0300
244.3270
1/12
22.0475
53.8861
93.2042
135.1220
177.5880
219.7490
1/10
21.6931
51.2278
85.8621
121.4510
156.8280
186.1870
50
0
23.4641
62.0769
121.1100
199.9840
298.6380
417.0390
1/20
23.1434
58.8234
108.4780
167.3280
231.8360
299.4830
1/18
23.0702
58.1314
106.0480
161.7410
221.7510
283.8340
1/16
22.9692
57.2037
102.9110
154.7970
209.6410
265.5900
1/14
22.8245
55.9270
98.7995
146.1120
195.0940
244.3780
1/12
22.6073
54.1176
93.3382
135.2140
177.6590
219.8170
1/10
22.2619
51.4712
86.0076
121.5530
156.8970
191.6860
100
0
24.5064
62.4783
121.3160
200.1090
298.7210
417.0980
1/20
24.1995
59.2469
108.7090
167.4780
231.9440
299.5660
1/18
24.1296
58.5599
106.2840
161.8960
221.8630
283.9220
1/16
24.0330
57.6391
103.1530
154.9580
209.7610
265.6840
1/14
23.8947
56.3722
99.0522
146.2830
195.2220
244.4800
1/12
23.6874
54.5776
93.6057
135.3990
177.8000
219.9290
1/10
23.3579
51.9546
86.2978
121.7590
157.0560
191.8160
200
0
26.4682
63.2735
121.7280
200.3590
298.8890
417.2180
1/20
26.1843
60.0849
109.1680
167.7760
232.1600
299.7330
1/18
26.1196
59.4076
106.7530
162.2040
222.0880
284.0980
1/16
26.0305
58.5001
103.6370
155.2810
209.9990
265.8730
1/14
25.9029
57.2523
99.5558
146.6250
195.4780
244.6850
1/12
25.7117
55.4862
94.1383
135.7680
178.0810
220.1820
1/10
25.4085
52.9083
86.8752
122.1690
157.3850
186.3180
Figure 4 illustrates how the first mode frequency (\(\Omega _1\)) of nanobeams under the SS, CC and CS boundary conditions varies with the elastic foundation stiffness parameter \(k\) and the nonlocal parameter \(\gamma /L\). Each graph highlights the impact of the elastic foundation on the natural frequencies of the system. As shown in Figure 4(a) (SS boundary condition) and Figure 4(b) (CC boundary condition), the first mode frequency increases as the foundation stiffness \(k\) increases. However, the rate of this increase is influenced by the \(\gamma /L\) ratio, with higher \(\gamma /L\) values generally resulting in lower frequencies. This indicates that nonlocal effects, particularly at higher \(\gamma /L\) values, introduce a noticeable softening effect. Furthermore, it is evident that the boundary condition plays a significant role, with the CC boundary condition showing the highest sensitivity to changes in both \(k\) and \(\gamma /L\).
Fig. 4
Influence of elastic foundation on the first mode frequencies varying with \(\gamma /L\) values for the SS, CC and CS boundary conditions
These behaviours can be explained by the fact that the elastic foundation stiffness increases the rigidity of the nanobeam, leading to higher vibration frequencies, while an increase in the nonlocal parameter \(\gamma /L\) causes the nanobeam to exhibit more flexible behaviour, resulting in lower frequencies. Additionally, the CC boundary condition provides the greatest constraint at the ends of the beam, leading to the highest frequencies. This highlights the critical role of both the elastic foundation stiffness and nonlocal effects in determining the dynamic behaviour of nanobeams. Figure 5 shows the variation of the first mode frequency \( \Omega _1 \) of nanobeams as a function of the non-dimensional nonlocal parameter \( \gamma /L \) for different elastic foundation stiffness values \( k \) under the SS, CC and CS boundary conditions. With an increase in the nonlocal parameter \( \gamma /L \), there is a consistent decrease in frequency \( \Omega _1 \) across all boundary conditions. Additionally, as the stiffness of the elastic foundation \( k \) increases, the frequency levels are elevated, with the highest frequencies observed under the CC boundary condition. The decrease in frequency with increasing \( \gamma /L \) suggests that nonlocal effects reduce the rigidity of the nanobeam, leading to lower natural frequencies. Conversely, the increase in frequency with higher \( k \) values reflects the supporting effect of a stiffer elastic foundation, which enhances the overall stiffness of the system.
Fig. 5
First mode frequency \( \Omega _1 \) of nanobeams versus the nonlocal parameter \( \gamma /L \) for different \( k \) values under various boundary conditions
It is important to note that the softening behaviour observed in the results of Eringen’s nonlocal model, particularly the reduction in natural frequencies with an increasing nonlocal parameter, is a characteristic feature of this formulation. However, recent studies, including experimentally calibrated stress-driven nonlocal models [83], have reported contrasting behaviour.
6 Conclusion
In this study, for the first time, a novel and effective approach was introduced to analyse the vibrational behaviour of SWCNTs on an elastic foundation. By combining the IVM and ATM within the framework of nonlocal elasticity theory, this approach offers an alternative to traditional methods for accurately capturing nanoscale dynamics, indicating that an increase in foundation stiffness significantly enhances natural frequencies across all boundary conditions. Conversely, higher nonlocal parameter values, representing nanoscale interactions, lead to a reduction in natural frequencies. This effect is particularly pronounced in higher vibrational modes. These results underscore the importance of accounting for both foundation stiffness and nonlocal effects in accurately analysing the dynamic behaviour of SWCNTs. Convergence analysis validates the computational efficiency and robustness of the methodology, confirming its capability for reliable vibrational predictions. Additionally, this methodological framework can serve as a versatile tool for analysing other nanostructures by incorporating different material properties, geometries and loading conditions. Future research could explore the integration of thermal effects into this framework to investigate their influence on vibrational characteristics and system stability. By addressing these factors, the proposed approach could contribute to the optimisation and design of nanoscale systems, including energy storage devices and nanoelectromechanical systems (NEMS).
Acknowledgements
The author sincerely thanks the editor and the anonymous reviewers for their constructive comments and valuable contributions, which have significantly improved the quality of the manuscript.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.
