Abstract
The double-nanobeam system has important applications in nano-optomechanical systems (NOMS), its dynamic analysis is of importance to the effective design of nanodevices. This paper aims to present analytical solutions of the forced vibration of a functionally graded double-nanobeam system (FGDNS) interconnected by a viscoelastic layer supported on an elastic foundation subjected to time-harmonic external forces. Employing the Hamilton’s principle, the governing differential equations of the FGDNS are derived in the context of the Euler–Bernoulli beam theory and Eringen’s nonlocal elasticity theory. Green’s functions method in conjunction with the superposition principle are adopted to obtain the explicit expressions of the steady-state responses of the FGNDS. A unified strategy applied to various boundary conditions is proposed to determine unknown constants involved in the Green’s functions. Meanwhile, the implicit equation calculating the natural frequency of the FGDNS is proposed. Numerical calculations are performed to check the validity of the present solutions and to discuss the influences of the small-scale parameter, material distribution parameter, and connecting layer parameters on dynamic behaviors of the FGNDS. Results show that the bond between the two nanobeams can be significantly reinforced by increasing the stiffness and damping coefficient of the connecting layer; the small-scale effect can soften or harden the system, depending upon the boundary conditions and the size of the frequency of external force.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11872319
Award Identifier / Grant number: 12072301
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Author contribution: Bo Chen: Conceptualization, Methodology, Writing-Original Draft, Software, Validation. Baichuan Lin: Writing-Review & Editing, Validation. Yukang Yang: Writing-Review & Editing, Validation. Xiang Zhao: Writing-Review & Editing, Funding acquisition. Yinghui Li: Methodology, Writing-Review & Editing, Supervision, Funding acquisition
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Research funding: This research work was supported by the National Natural Science Foundation of China (Grant Nos. 11872319 and 12072301).
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Conflict of interest statement: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The explicit expressions of ϕ ji (j = 1, 2, …, 8; i = 1, 2) in Eqs. (45) and (46) are given by
where
in which, s i (i = 1,2, …, 8) are eight roots of the following polynomial equation:
The explicit expressions of the kth (k = 1, 2, 3)-order derivative of ϕ ji (j = 1, 2, …, 9; i = 1, 2) in Eqs. (47) and (48) are given by
As can be seen, the influence of the connecting layer damping coefficient on the steady-state responses is discussed separately in the Subsection 4.4. This is because, when the damping factor is considered, the vibrations of arbitrary two points on the system are not synchronous, there is a phase difference in two points. This fact implies that the dynamic deflections of the system considering the damping effect are not separable in time and space, and the corresponding mathematical proof in detail is provided as follows.
From Eqs. (32) and (33), it is known that the coefficients a m and b m (m = 1, 2, 3, 4) are the complex numbers if the damping coefficient c d ≠ 0. Thus, the solutions to Eqs. (30) and (31) may can be written as
in which both α j and β j are real functions with respect to x. It is worth pointing out that, for an undamped system, namely c d = 0, we have β j (x) = 0. Then, the dynamic deflections w j (x,t) ( j = 1,2) of the system under the time-harmonic external force p(x,t) = P(x)eiΩt are obtained as [45]:
if p(x,t) = P(x)cos(Ωt), we get
(B.2)with
(B.3)if p(x,t) = P(x)sin(Ωt), we get
(B.4)with
(B.5)
According to Eqs. (B.2)B.5)–(B.5), it can be known that:
Obviously, when the damping coefficient c d = 0, the phase difference φ j (x) is equal to zero. According to Eq. (B.2) or (B.4), the dynamic deflections of the system are separable in time and space, namely, w j (x,t) = W j (x)cos(Ωt) or W j (x)sin(Ωt). In this case, W j (x) not only reflect the deflection amplitudes of the system but also the vibration shape, and the time term (cos(Ωt) or sin(Ωt)) is equivalent to a scale factor for the system’s dynamic deflections. Thus, the steady-state deflection amplitudes W j (x) of the system are enough to show its dynamic behaviors when the damping effect of the connecting layer is not considered, as can be seen in the Subsections 4.2 and 4.3.
Further, when the damping coefficient c d ≠ 0, the dynamic deflections of the system at the arbitrary two points x = x1 and x = x2 are not synchronous, having the phase difference |φ j (x1) − φ j (x2)|. In other word, the dynamic deflections of the system considering the damping factor are not separable in time and space, namely, cannot put in the form w j (x,t) = W(x)q(t) due to φ j (x) ≠ 0. Thus, we have to present the dynamic deflection w j (x,t) of the system, instead of the steady-state deflection amplitudes W j (x), to show its dynamic behaviors, as can be seen in the Subsection 4.4.
Based on the above analyses, especially 2nd and 3rd bullet points, for simplicity, the authors choose to separately investigate the influence of the damping coefficient of the connecting layer on steady-state responses of the FGNDS in the Subsection 4.4.
Based on the Eringen’s nonlocal elasticity theory, Murmu and Adhikari [6] investigated the free vibration of a simply supported double-nanobeam system interconnect by the Winkler elastic layer made by homogeneous materials shown in Figure C.1, with no consideration supported by foundation. The length of the two nanobeams is L. The flexural rigidity, mass of per unit length, scale parameter, and axial load of the two nanobeams are considered to be identical, and denoted by EI, m, e0a, and N, respectively; the stiffness of the connecting layer is denoted by k c .
A simply supported double-nanobeam system under axial load.
Using their research, the vibration of a double-nanobeam system may have two patterns, in-phase or out-of-phase, depending on whether the two sub-nanobeams move toward the same direction, as can be seen in Figure C.2, and the nth-order dimensionless natural frequency of two vibration patterns are determined by the following expressions [6].
Two vibration patterns of the double-nanobeam system.
In-phase vibration:
Out-of-phase vibration:
in which
where Ω represent the natural frequency of the system. Note that the nth-order dimensionless natural frequency of in-phase/out-of-phase vibration pattern should be the (2n − 1)th/(2n)th order dimensionless natural frequency of this system. Data for the numerical example used in Section 4.1 is Γ = 0.5.
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