Royal Society Publishing

Free transverse vibrations of nano-to-micron scale beams

Mingtian Xu

Abstract

In the present work, the integral equation approach and the non-local elasticity theory are employed to investigate the free transverse vibrations of nano-to-micron scale beams. The frequency equation is analytically formulated into an eigenvalue problem of a matrix with an infinite order. The numerical calculation is implemented by truncating this matrix to a finite order one. It is found that the impact of the non-local effect on the natural frequencies and vibrating modes is negligible for the beams with micrometre scale length. But when the length of beams reaches the nanoscale, the non-local effect becomes important, especially for the high-order natural frequencies and vibrating modes.

Keywords:

1. Introduction

The rapidly developing nanotechnology brings us more and more microelectromechanical devices. The dimension of these devices now can reach nanometre scale. Owing to the requirements in applications, the mechanical properties of these nanoscale devices have drawn tremendous attention. Two approaches are available for such investigations: the classical continuum mechanics and the atomic or molecular models. Both have some drawbacks. There is evidence that the applicability of classical continuum mechanics to nanoscale devices is doubted (Broughton et al. 1997; Carr et al. 1999; Rudd & Broughton 1999). The difference of the natural frequencies predicted by the continuum elasticity theory and the molecular simulation can reach 10–38% for a 17 nm scale plate with the clamped ends (Rudd & Broughton 1999). Even for a sub-micron length-scale quartz crystal oscillator, the surface effect and the anharmonicity phenomenon may occur which is unexplainable by the continuum elasticity theory (Carr et al. 1999). Furthermore, for a sub-micron scale mechanical resonator, it is possible to observe quantum effects under certain conditions (Blencowe 2004). Atomic or molecular simulation is becoming a powerful tool to explore the properties of nanodevices owing to the rapidly growing capacity of the supercomputer and Beowulf clusters technology. But it is difficult to establish and the computational cost are also expensive. Therefore, the non-local continuum mechanics (Eringen 1972a,b, 1983, 1992; Eringen et al. 1977), which is the improvement of the classical continuum mechanics by counting the impact of the microstructure of materials, provides us with a new tool to deal with these tiny devices in a trade-off way (Peddieson et al. 2003).

The beam-like microscale devices are key components in microelectromechanical systems, such as cantilever actuators, nanotubes, nanowires, the nano-to-micron scale mechanical resonator in quantum electromechanical system and microcantilever-tip systems in atomic force microscopy. The thorough understanding of the vibrational behaviour of these structures is of great practical and theoretical interest. The vibrational property of the carbon nanotube has already been employed to study its elastic bending modulus (Poncharal et al. 1999). The vibration analysis of nanowires is required in order to determine the lattice thermal conductance (Blencowe 2004; Tanaka et al. 2005). The analysis of the forced vibration of the cantilever-tip system is fundamental to the atomic force microscopy, which is a powerful and versatile technique for atomic and nanoscale tomography and manipulations of surfaces, DNA, proteins and polymers (Garcia & Perez 2002; Ward 2005). Usually, the classical Bernoulli/Euler beam model is employed to describe the vibration of these nano-to-micron scale beam structures (Treacy et al. 1996; Poncharal et al. 1999; Kahn et al. 2001; Yong et al. 2002; Xia et al. 2004; Tanaka et al. 2005). However, the applicability of the classical continuum models for these systems is called into question (Ruoff et al. 1993; Lu 1997; Hertel et al. 1998; Govindjee & Sackman 1999). Recently, the non-local elasticity theory has been developed to investigate the bending of nano-to-micron scale beams under static loads (Peddieson et al. 2003) and the free transverse vibrations of carbon nanotubes (Zhang et al. 2005). In the present work, we focus on the free vibration of the general nano-to-micron scale beams based on the non-local continuum models.

2. Non-local continuum beam model

In classical continuum elasticity, the stress state at a point x is only related to the strain state at the same point x. This view contradicts the atomic theory of lattice mechanics and experimental observations of phonon dispersion. In order to resolve this paradox, non-local continuum mechanics has been put forward by Eringen et al. (Eringen 1972a,b, 1983, 1992; Eringen et al. 1977). In this theory, the stress state at a point x is relevant to all the points of the body. For the linear and isotropic elastic solids, the non-local elasticity theory is mathematically formulated as follows (Eringen 1992):Embedded Image(2.1)Embedded Image(2.2)Embedded Image(2.3)where Embedded Image, Embedded Image, ρ, Embedded Image and Embedded Image are the non-local stress tensor, the classical stress tensor, mass density, body force density and the displacement vector at a point Embedded Image in the body, respectively. λ and μ are the Lamé constants. V is the volume occupied by the elastic body. The non-local kernel Embedded Image reflects the impact of the strain at the point Embedded Image on the stress at the point Embedded Image. Note that the constitutive equation in the non-local continuum elasticity is expressed by an integral over the entire elastic body. Actually, it reveals the averaged influence of the strain at all the points in the body on the stress at the point Embedded Image. Compared with the classical continuum elasticity, the governing equation of the non-local continuum elasticity is quite complicated. Fortunately, under certain conditions, it can be simplified by converting the integral constitutive equation in equation (2.1) to the equivalent differential constitutive equation. For example, in the non-local elasticity, the uniaxial Hooke's law is expressed as (Eringen 1983; Peddieson et al. 2003; Zhang et al. 2005)Embedded Image(2.4)where σ is the axial stress, ϵ the axial strain, E the Young modulus, a the internal characteristic length (length of C–C bond, lattice spacing, granular distance), Embedded Image a constant to be determined for each material and x the axial coordinate.

For the transversely vibrating beam, the equilibrium conditions read (Bishop & Johnson 1979; Peddieson et al. 2003; Zhang et al. 2005)Embedded Image(2.5)Embedded Image(2.6)where S is the shear force, M the bending moment, p the loading per unit length, ρ the mass density, A the area of the cross-section of the beam, w the beam's transverse displacement and t the time variable. Combining equations (2.5) and (2.6) yieldsEmbedded Image(2.7)Multiplying by y on both sides of equation (2.4) and integrating over the cross-section of the beam at the point x, we obtainEmbedded Image(2.8)According to the definition of the bending moment, we haveEmbedded Image(2.9)The small deflection Bernoulli/Euler relation between strain and curvature isEmbedded Image(2.10)For the uniform beam, substituting equations (2.9) and (2.10) into equation (2.8) yieldsEmbedded Image(2.11)where I is the area moment of inertia. Differentiating both sides of this equation with respect to the variable x twice givesEmbedded Image(2.12)Substituting equation (2.7) into the above equation yieldsEmbedded Image(2.13)This is the governing equation of beams based on the non-local continuum elasticity (Zhang et al. 2005). In addition, Embedded Image should satisfy appropriate boundary conditions.

In the following, four kinds of boundary conditions are considered.

  • For the simply supported beam, its boundary condition is written asEmbedded Image(2.14)

  • For the clamped–clamped beam,Embedded Image(2.15)

  • For the clamped–hinged beam,Embedded Image(2.16)

  • For the cantilever beam,Embedded Image(2.17)

Substituting equation (2.7) into equation (2.11) with the assumption Embedded Image yieldsEmbedded Image(2.18)An application of equation (2.5) givesEmbedded Image(2.19)Note that by setting a=0 in equation (2.13), we obtain the classical Bernoulli/Euler beam equation. Setting Embedded Image (L is the length of the beam) in equation (2.13) and still denoting Embedded Image as x, givesEmbedded Image(2.20)Since in the following we only address the free vibration of the beam, p can be set to zero and Embedded Image be written asEmbedded Image(2.21)where Embedded Image and ω is the natural frequency. Therefore, equation (2.20) becomesEmbedded Image(2.22)where Embedded Image and Embedded Image. Equations (2.18) and (2.19) arrive atEmbedded Image(2.23)Embedded Image(2.24)where the prime symbol represents the derivative with respect to the variable x.

Subsequently, equations (2.14)–(2.17) become:

  • for the simply supported beam,Embedded Image(2.25)

  • for the clamped–clamped beam,Embedded Image(2.26)

  • for the clamped–hinged beam,Embedded Image(2.27)

  • for the cantilever beam,Embedded Image(2.28)

Obviously, these boundary conditions can be reduced to:

  • for the simply supported beam,Embedded Image(2.29)

  • for the clamped–clamped beam,Embedded Image(2.30)

  • for the clamped–hinged beam,Embedded Image(2.31)

  • for the cantilever beam,Embedded Image(2.32)

Therefore, when the non-local effect is counted, only the boundary condition of the cantilever beam has been changed; the others considered in the present work are the same as that for the classical Bernoulli/Euler beam theory.

3. Free vibration of nano-to-micron beams

In this section, the natural frequencies and vibrating modes of the nano-to-micron beams are investigated by the integral equation approach (Xu & Cheng 1994; Xu et al. 1998). By applying this approach, the free transverse vibration of beams can be analytically formulated to a linear eigenvalue problem of a matrix with an infinite order, which can be solved numerically by truncating this matrix to a finite order. Surprisingly, the truncated second-order matrix already gives us a satisfactory result. Therefore, an approximate formula of the fundamental frequency can be obtained. In the following, this method will be exemplified by the cantilever beam.

In order to convert equation (2.22) with the boundary condition (2.32) into an equivalent integral equation form, we first consider the following auxiliary differential equationEmbedded Image(3.1)Embedded Image(3.2)where Embedded Image is Dirac δ-function. Note that Embedded Image is nothing but the Green function of the cantilever beam with Embedded Image. By equation (A 6) in appendix A and the boundary condition (3.2), Embedded Image can be written asEmbedded Image(3.3)where Embedded Image is the third-order derivative of G at the end x=0, Embedded Image is the rotation angle at the end x=1. Substituting the cosine series of δ-function,Embedded Imageand equation (3.3) into equation (3.1), yieldsEmbedded Image(3.4)The application of the boundary condition Embedded Image givesEmbedded Image(3.5)Substituting equation (3.5) into equation (3.3) yieldsEmbedded Image(3.6)By the superposition principle, equation (2.22) with the boundary condition (2.32) can be transformed into the following integral equationEmbedded Image(3.7)The proof of this statement is given in appendix B. Substituting equation (3.6) into the above equation and denotingEmbedded Image(3.8)Embedded Image(3.9)we haveEmbedded Image(3.10)Differentiating both sides of this equation givesEmbedded Image(3.11)Embedded Image(3.12)Putting equations (3.10) and (3.12) into equations (3.8) and (3.9) and applying the following property of the Delta functionEmbedded Image(3.13)Embedded Image(3.14)where Embedded Image, we obtainEmbedded Image(3.15)Embedded Image(3.16)These equations can be expressed as an eigenvalue problem of a matrix,Embedded Image(3.17)withEmbedded ImageEmbedded ImageEmbedded ImageEmbedded ImageEmbedded ImageSimilarly, using the sine Fourier series expansion (A 5) in appendix A, the frequency equations of other kinds of beams with different boundary conditions can also be obtained. They have the same form as equation (3.17). But the elements of the matrix Embedded Image are different. For the clamped–clamped and clamped–hinged beams, the results are listed in the following.

  • Clamped–clamped beam:Embedded Image(3.18)

  • Clamped–hinged beam:Embedded Image(3.19)

For the simply supported beam, the exact natural frequencies and vibrating modes can be obtained,Embedded Image(3.20)where Embedded Image is the nth vibrating mode. For other beams, we have to truncate the matrix Embedded Image into a finite-order matrix in order to implement the numerical calculation. Then, the QR method is employed to solve this eigenvalue problem. By the obtained eigenvalues and eigenfunctions, we can get the natural frequencies and the vibrating modes. From table 1 one can see that a good convergence of this integral equation method has been achieved for the classical Bernoulli/Euler beam. Actually, the truncated second-order matrix Embedded Image already gives us a satisfactory result. Therefore, an approximate formula for the fundamental frequency of these beams can be obtained,Embedded Image(3.21)where Embedded Image

View this table:
Table 1

The convergence of the fundamental frequency for the integral equation approach.

4. Results

By the method described earlier, one can investigate the influence of the non-local effect on the frequencies and vibrating modes of the nano-to-micron cantilever beam, simple supported beam, clamped–clamped beam and the clamped–hinged beam. Concerning the importance of the low-order modes, we only present the first three natural frequencies and the first two vibrating modes in this paper. In order to do the calculation, we have to determine which value of Embedded Image should be used. In Zhang et al. (2005), by the molecular simulation result of carbon nanotubes, it has been found that Embedded Image approximately equals 0.82. Such a value of Embedded Image is assumed in the following calculations. The impact of Embedded Image on the natural frequencies of the simply supported beam and the cantilever beam with a length Embedded Image is displayed in table 2. For other kinds of beams considered in this paper, the variation of the natural frequencies with respect to Embedded Image is quite similar to the simply supported beam case.

View this table:
Table 2

The influence of the parameter Embedded Image on the natural frequencies of the simply supported beam and cantilever beam with the length L=60a, which is expressed by equation (4.1).

Tables 3–6 show the impact of the non-local effect on the natural frequencies of beams with different lengths. Such an impact is quantified asEmbedded Image(4.1)where ω is the frequency of the classical beam, while Embedded Image is the frequency with the non-local effect. It increases the fundamental and second frequencies of the cantilever beam, but decreases the natural frequencies for all other cases. In Carr et al. (1999), the vibration of a 2 μm long, Embedded Image thick and Embedded Image wide clamped–clamped beam was investigated experimentally. A fundamental frequency of Embedded Image has been measured. Based on the classic Bernoulli/Euler beam, the fundamental frequency is Embedded Image. By taking into account the non-local effect, our calculated results in table 4 show that the fundamental frequency of the clamped–clamped beam should be slightly less than Embedded Image; therefore, it is little more accurate compared to the experimental result. In Rudd & Broughton (1999), it was found that the fundamental frequency obtained by molecular simulation is always lesser than that obtained by the classical continuum elasticity theory for a Embedded Image long plate with clamped ends. This tendency is also consistent with our finding for the clamped–clamped beam. In future work, we will apply our method to investigate the vibration of carbon nanotubes and compare with the experimental or molecular simulation results.

View this table:
Table 3

The influence of the non-local effect on the natural frequencies of the simply supported beam, which is expressed by equation (4.1).

View this table:
Table 4

The influence of the non-local effect on the natural frequencies of the clamped–clamped supported beam, which is expressed by equation (4.1).

View this table:
Table 5

The influence of the non-local effect on the natural frequencies of the clamped–hinged supported beam, which is expressed by equation (4.1).

View this table:
Table 6

The influence of the non-local effect on the natural frequencies of the cantilever beam, which is expressed by equation (4.1).

For the simply supported beam, equation (3.20) indicates that the non-local effect has no influence on the vibrating modes. For other kinds of beams, the impact of the non-local effect on the vibrating modes is displayed in figures 1–6. In order to clearly illustrate such an impact: when plotting these figures, we first discretize the interval Embedded Image into 100 subintervals, by the points Embedded Image. Then, the influence of the non-local effect on the vibrating modes at the point Embedded Image is quantified asEmbedded Image(4.2)where Embedded Image and Embedded Image are the transverse displacements of one mode at the point Embedded Image without and with the non-local effect, respectively. From these figures, one can see that the higher the order of vibrating modes, the greater the impact of the non-local effects. But when the length of the nano-to-micron beam is in the micrometre scale, the non-local effect has slight impact on both the natural frequencies and the vibrating modes.

Figure 1

The influence of the non-local effect on the first mode of the cantilever beam, which is expressed by equation (4.2). The solid, dashed, dash-dot and dash-dot-dot lines are for beams with length Embedded Image, 60a, 40a and 20a, respectively.

Figure 2

The influence of the non-local effect on the second mode of the cantilever beam, which is expressed by equation (4.2). The solid, dashed, dash-dot and dash-dot-dot lines are for beams with the length Embedded Image, 60a, 40a and 20a, respectively.

Figure 3

The influence of the non-local effect on the first mode of the clamped–clamped beam, which is expressed by equation (4.2). The solid, dashed, dash-dot and dash-dot-dot lines are for beams with the length Embedded Image, 60a, 40a and 20a, respectively.

Figure 4

The influence of the non-local effect on the second mode of the clamped–clamped beam, which is expressed by equation (4.2). The solid, dashed, dash-dot and dash-dot-dot lines are for beams with the length Embedded Image, 60a, 40a and 20a, respectively.

Figure 5

The influence of the non-local effect on the first mode of the clamped–hinged beam, which is expressed by equation (4.2). The solid, dashed, dash-dot and dash-dot-dot lines are for beams with the length Embedded Image, 60a, 40a and 20a, respectively.

Figure 6

The influence of the non-local effect on the second mode of the clamped–hinged beam, which is expressed by equation (4.2). The solid, dashed, dash-dot and dash-dot-dot lines are for beams with the length Embedded Image, 60a, 40a and 20a, respectively.

5. Concluding remarks

The integral equation approach is employed to examine the influence of the non-local effect on the transverse vibrations of the nano-to-micron beams. It is found that for the beams with the micrometre scale length, the classical Bernoulli/Euler beam theory is applicable; but for the beams with nanometre scale length, the influence of the non-local effect becomes larger, especially for the high-order frequencies and vibrating modes. Interestingly, for the cantilever beam, it demonstrates a quite different behaviour for its first and second modes.

In the present work, although the non-local effect has been taken into account, it is still a quite simple model of the nano-to-micron beam. Many important factors for the nano-micron-scale beams have not been addressed, for example, the size-dependence of the elastic properties (Miller & Shenoy 2000), the energy losses due to the surface-related effects (Carr et al. 1999), the temperature dependency and quantum effects (Blencowe 2004), etc. These will be left for future work.

Acknowledgements

I am indebted to the referees for their invaluable suggestions.

Footnotes

    • Received December 1, 2005.
    • Accepted March 6, 2006.

References

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