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Surface effects on the quasi-periodical free vibration of the nanobeam: semi-analytical solution based on the residue harmonic balance method

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Abstract

The study of surface effects on quasi-periodical vibration is significant because of its promising applications in sensors of nano/micro-electro-mechanical systems (N/MEMS). In this study, the surface effects parameters are adopted from the atomistic calculations. The governing equation of the nanobeam considering the nonlinear curvature and surface effects is established. Considering the first two modes, the Garlerkin method is used to translate the partial differential equation into ordinary differential equations. The multi-level residue harmonic balance method (RHBM) with two time variables is developed to solve these ODEs. The influences of surface and aspect ratio on the two frequencies and corresponding amplitudes are analyzed. Both of the frequency and amplitude discrepancies of the nanobeam to the counterparts of the classical beam become greater when the height of the beam shrinks at nanoscale. For the beam with the same height, both of the first- and second-frequencies decrease with the increase of the aspect ratio, and for the beams with the same aspect ratio, they decrease with the increase of the height. The solution of RHBM fits well with the molecular dynamics and Runge–Kutta numerical simulations. The study provides insight into the mechanism of the nonlinear dynamics of nanostructures, and shed light on quantitative design of the elements in N/MEMS, sensors, actuators, and resonators.

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Funding

This study is supported by the National Natural Science Foundation of China (Grand Number 11672334 and 11672335), and Key Research and Development Province (Grand Number 2017GGX20117).

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Correspondence to Demin Zhao or Jianlin Liu.

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Appendices

Appendix A

Formulation the surface effect based on the surface elasticity theorem in Sect. 2.

The surface stress tensor \({\varvec{\uptau}}_{\alpha \beta }\) within the surface layer, which is related to surface energy density \(\gamma\) and strain \({\varvec{\upvarepsilon}}_{\alpha \beta }\),can be expressed by \({\varvec{\uptau}}_{\alpha \beta } \left( \varepsilon \right) = \gamma {\varvec{\updelta}}_{\alpha \beta } + \frac{\partial \gamma }{{\partial {\varvec{\upvarepsilon}}_{\alpha \beta } }}\), where the \(\delta_{\alpha \beta } { = 1}\) for \(\alpha { = }\beta\) and \(\delta_{\alpha \beta } { = }0\) for \(\alpha \ne \beta\). For one-dimension nanobeam, surface stress can be reduced into \(\tau \left( \varepsilon \right){ = }\tau^{s} + E^{s} \varepsilon\), where \(\tau^{s}\) is residue surface tension of the surface at zero strain and Es is the surface Young modulus. The influence of longitudinal stress on the elastic bending behavior of nanobeam is assumed to be much smaller than the distributed transverse force. The transverse stress, acting on the upper and the lower surfaces of the beam \(\sigma_{ij}^{ \pm }\) results from the in-plane stress \(\tau^{s}\), can be described by the generalized Young–Laplace equation. The jump of the transverse stress \(\left\langle {\sigma_{ij}^{ + } - \sigma_{ij}^{ - } } \right\rangle n_{i} n_{j} = \tau^{s} \kappa\). where \(\kappa\) is principle curvature and the ni is the unit vector normal the surface. Because the beam is very thin, the magnitudes of \(\sigma_{ij}^{ + }\) and \(\sigma_{ij}^{ - }\) are same. Therefore, the transverse stress induce the surface residue stress is given by \(q_{s} = H^{s} \kappa\), where \(H^{s} = 2\tau^{s} b\).

Based on the theorem of composite beam, the effect of the surface elasticity Es on the flexural rigidity for a rectangular cross section beam is given by \(\left( {EI} \right)^{*} = \frac{{Ebh^{3} }}{12} + \frac{{E^{s} h^{3} }}{6} + \frac{{E^{s} bh^{2} }}{2}\).

Next we introduce the derivative process of the government equation. Assume the cantilever beam is inextensible, then the derivatives of x and W with respective s are is defined as

$$x^{\prime} = \frac{\partial x}{\partial s},\;W^{\prime} = \frac{\partial W}{\partial s}$$

and the following equations are obtained

$$W^{\prime 2} + x^{\prime 2} = 1,\;W^{\prime } = \sqrt {1 - x^{\prime 2} } ,$$

The relationship of \(\delta x\) and \(\delta W\) is derived as following

$$\begin{aligned} \delta x^{\prime } & = - \frac{{W^{\prime } {\kern 1pt} \delta W}}{{\sqrt {1 - W^{\prime 2} } }} = - W^{\prime } \left( {1 + \frac{1}{2}W^{\prime 2} } \right)\delta W, \\ \delta x & = \int_{0}^{s} { - \frac{{W^{\prime } {\kern 1pt} \delta W}}{{\sqrt {1 - W^{\prime 2} } }}} {\text{d}}s = - W^{\prime } \left( {1 + \frac{1}{2}W^{\prime 3} } \right)\delta W + \int_{0}^{s} {\left( {W^{\prime \prime } + \frac{3}{2}W^{\prime 2} W^{\prime \prime } } \right)\delta W{\kern 1pt} {\text{d}}s} \\ & \approx - W^{\prime } {\kern 1pt} \delta W{ + }\int_{0}^{s} {W^{\prime \prime } {\kern 1pt} \delta W{\kern 1pt} {\text{d}}s} . \\ \end{aligned}$$

The curvature of the beam \(\kappa\) and the square of the curvature \(\kappa^{2}\) are given by

$$\kappa = \frac{{W^{\prime \prime } }}{{\sqrt {1 - W^{\prime 2} } }} = W^{\prime \prime } \left( {1{ + }\frac{{W^{\prime 2} }}{2}} \right){\kern 1pt} ,\;\;\kappa^{2} { = }W^{\prime \prime 2} \left( {1 - W^{\prime 2} } \right)^{ - 1} { = }W^{\prime \prime 2} \left( {1{ + }W^{\prime 2} } \right).$$

The strain and the kinetic energies can be expressed as

$$U = \frac{EI}{2}\int_{0}^{L} {\kappa^{2} } {\text{d}}s = \frac{EI}{2}\int_{0}^{L} {W^{\prime \prime 2} \left( {1{ + }W^{\prime 2} } \right)} {\text{d}}s,\;{\text{and}}\;T = \frac{m}{2}\int_{0}^{L} {(\dot{W}^{2} } + \dot{x}^{2} ){\text{d}}s.$$

Based on Hamilton principle

$$\delta \int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\left( {T - U} \right){\text{d}}\bar{t}} + \int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\delta W_{e} {\text{d}}\bar{t}} = 0,$$

where \(\delta W_{e}\) is the virtual work of the non-conservative forces. The variation of the virtual work can be written as

$$\int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\delta W_{e} } {\text{d}}\bar{t} = \int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\int_{0}^{L} {q_{s} \delta W{\kern 1pt} {\text{d}}s{\text{d}}\bar{t} = } } H^{s} \int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\int_{0}^{L} {W^{\prime \prime } \left( {1 + \frac{1}{2}W^{\prime 2} } \right)\delta W{\kern 1pt} {\text{d}}s} } {\text{d}}\bar{t}.$$

The variation of the strain and the kinetic energies can be written as

$$\begin{aligned} \delta U & = \frac{EI}{2}\delta \int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\int_{0}^{L} {\kappa^{2} } {\text{d}}s{\kern 1pt} } {\text{d}}\bar{t} = - EI\int_{0}^{L} {\left( {W^{\prime \prime \prime \prime } + W^{\prime \prime \prime \prime } W^{\prime 2} + 4W^{\prime } W^{\prime \prime } W^{\prime \prime \prime \prime } + W^{\prime \prime 2} } \right)} \delta W{\kern 1pt} {\text{d}}s{\text{d}}\bar{t}; \\ \delta T & = \frac{m}{2}\delta \int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\int_{0}^{L} {(\dot{W}^{2} } + \dot{x}^{2} ){\text{d}}s} {\kern 1pt} {\kern 1pt} {\text{d}}\bar{t} = - m\int_{{\bar{t}_{1} }}^{{\bar{t}_{2} }} {\int_{0}^{L} {\left( {\ddot{W}{\kern 1pt} \delta W + \ddot{x}{\kern 1pt} \delta x} \right){\kern 1pt} {\kern 1pt} } {\text{d}}s\,} {\text{d}}\bar{t}. \\ \end{aligned}$$

Derivative both sides of the equation \(W^{\prime } + x^{\prime } = 1\) with respective to time \(\bar{t}\) yields \(2W^{\prime } \dot{W}^{\prime } + 2x^{\prime } \dot{x}^{\prime } = 0\). Therefore we obtain \(\dot{x}^{\prime} = - \frac{{\left( {{{\partial W} \mathord{\left/ {\vphantom {{\partial W} {\partial s}}} \right. \kern-0pt} {\partial s}}} \right)\dot{W}^{\prime}}}{{\left( {{{\partial x} \mathord{\left/ {\vphantom {{\partial x} {\partial s}}} \right. \kern-0pt} {\partial s}}} \right)}} = - W^{\prime}{\kern 1pt} \dot{W}^{\prime}\).

The velocity and acceleration of the beam along x direction

$$\dot{x} = \int_{0}^{s} { - W^{\prime } \dot{W}^{\prime } } {\text{d}}s;\;\ddot{x} = - \int_{0}^{s} {\left( {\dot{W}^{\prime 2} + W^{\prime } \ddot{W}^{\prime } } \right)} {\kern 1pt} {\text{d}}s$$

Followed the above steps, the government equation Eq. (1) can be derived.

Appendix B

In Sect. 2, the parameters ki1, ki2, ki3, ki4, ki5, ki6, ci1, ci2, ci3, ci4, ci5, ci6 (i = 1, 2) of Eqs. (7) and (8).

$$\begin{aligned} k_{11} & = 12.3596 - 0.8588\beta + 12.3596\mu ;\;\;k_{12} = 0.0068 + 11.7444\beta + 0.0068\mu ; \\ k_{13} & = - 16.3892 - 2.1571\beta + 40.4251\mu ;\;\;k_{14} = 1528.0120 + 41.2266\beta - 306.8542\mu ; \\ k_{15} & = - 12783.2313 - 147.2032\beta + 2483.9228\mu ;\;\;k_{16} = 8597.5497 + 192.7130\beta - 2176.2470\mu ; \\ c_{11} & = \, 4.5968;\;\;c_{12} = - 3.5964;\;\;c_{13} = - 7.1928;\;\;c_{14} = 12.2359;\;\;c_{15} = 25.1741;\;\;c_{16} = - 22.1929; \\ k_{21} & = 0.0002 - 1.8739\beta + 0.0002\mu ;\;\;k_{22} = 485.4811 + 13.2938\beta + 485.4811\mu ; \\ k_{23} & = - 57.8499 - 3.4640\beta - 102.2985\mu ;\;\;k_{24} = - 872.6730 - 32.0639\beta + 2484.1643\mu ; \\ k_{25} & = - 15292.1786 - 44.4417\beta - 6530.1215\mu ;\;\;k_{26} = - 56843.5188 + 53.2269\beta + 13415.5523\mu ; \\ c_{21} & = - 3.5963;\;\;c_{22} = 25.1733;\;\;c_{23} = 12.2356;\;\;c_{24} = - 44.3844;\;\;c_{25} = 22.1922;\;\;c_{26} = 144.7195. \\ \end{aligned}$$

Appendix C

In Sect. 3.1, the Garlerkin procedure to obtain the parameters \(\Delta_{i,1}\) and \(\Delta_{i,2}\) (i = 1, 2) in Eqs. (22) and (23)

$$\begin{aligned} \Delta_{i,1} \left( {\omega_{1,0} ,\omega_{2,0} ,a_{1,0} ,a_{2,0} ,b_{1,0} ,b_{2,0} } \right) = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 0 \right)} \cos \tau_{1} } } {\text{d}}\tau_{1} {\text{d}}\tau_{2} ; \hfill \\ \Delta_{i,2} \left( {\omega_{1,0} ,\omega_{2,0} ,a_{1,0} ,a_{2,0} ,b_{1,0} ,b_{2,0} } \right) = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 0 \right)} \cos \tau_{2} } } {\text{d}}\tau_{1} {\text{d}}\tau_{2} . \hfill \\ \end{aligned}$$

The Garlerkin procedure to obtain the parameters \(\Delta_{i,1j}\), (i = 1, 2, j = 1,2,…,7) in Eqs. (32)

$$\begin{aligned} & \Delta_{i,11} \left( {\omega_{1,1} ,\omega_{2,1} ,a_{1,11} ,b_{1,11} ,a_{1,12} ,b_{1,12} ,a_{1,13} ,b_{1,13} ,a_{2,11} ,b_{2,11} ,a_{2,12} ,b_{2,12} ,a_{2,13} ,b_{2,13} } \right) \\ & \quad = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 1 \right)} \cos \tau_{1} } } {\text{d}}\tau_{1} {\text{d}}\tau_{2} ; \\ \end{aligned}$$
$$\begin{aligned} & \Delta_{i,12} \left( {\omega_{1,1} ,\omega_{2,1} ,a_{1,11} ,b_{1,11} ,a_{1,12} ,b_{1,12} ,a_{1,13} ,b_{1,13} ,a_{2,11} ,b_{2,11} ,a_{2,12} ,b_{2,12} ,a_{2,13} ,b_{2,13} } \right) \\ & \quad = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 1 \right)} \cos \tau_{2} } } {\text{d}}\tau_{1} {\text{d}}\tau_{2} ; \\ \end{aligned}$$
$$\begin{aligned} & \Delta_{i,13} \left( {\omega_{1,1} ,\omega_{2,1} ,a_{1,11} ,b_{1,11} ,a_{1,12} ,b_{1,12} ,a_{1,13} ,b_{1,13} ,a_{2,11} ,b_{2,11} ,a_{2,12} ,b_{2,12} ,a_{2,13} ,b_{2,13} } \right) \\ & \quad = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 1 \right)} \cos \left( {3\tau_{1} } \right)} } {\text{d}}\tau_{1} {\text{d}}\tau_{2} ; \\ \end{aligned}$$
$$\begin{aligned} & \Delta_{i,14} \left( {\omega_{1,1} ,\omega_{2,1} ,a_{1,11} ,b_{1,11} ,a_{1,12} ,b_{1,12} ,a_{1,13} ,b_{1,13} ,a_{2,11} ,b_{2,11} ,a_{2,12} ,b_{2,12} ,a_{2,13} ,b_{2,13} } \right) \\ & \quad = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 1 \right)} \cos \left( {2\tau_{2} + \tau_{1} } \right)} } {\text{d}}\tau_{1} {\text{d}}\tau_{2} ; \\ \end{aligned}$$
$$\begin{aligned} & \Delta_{i,15} \left( {\omega_{1,1} ,\omega_{2,1} ,a_{1,11} ,b_{1,11} ,a_{1,12} ,b_{1,12} ,a_{1,13} ,b_{1,13} ,a_{2,11} ,b_{2,11} ,a_{2,12} ,b_{2,12} ,a_{2,13} ,b_{2,13} } \right) \\ & \quad = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 1 \right)} \cos \left| {\tau_{2} - 2\tau_{1} } \right|} } {\text{d}}\tau_{1} {\text{d}}\tau_{2} ; \\ \end{aligned}$$
$$\begin{aligned} & \Delta_{i,16} \left( {\omega_{1,1} ,\omega_{2,1} ,a_{1,11} ,b_{1,11} ,a_{1,12} ,b_{1,12} ,a_{1,13} ,b_{1,13} ,a_{2,11} ,b_{2,11} ,a_{2,12} ,b_{2,12} ,a_{2,13} ,b_{2,13} } \right) \\ & \quad = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 1 \right)} \cos \left( {\tau_{2} + 2\tau_{1} } \right)} } {\text{d}}\tau_{1} {\text{d}}\tau_{2} ; \\ \end{aligned}$$
$$\begin{aligned} & \Delta_{i,17} \left( {\omega_{1,1} ,\omega_{2,1} ,a_{1,11} ,b_{1,11} ,a_{1,12} ,b_{1,12} ,a_{1,13} ,b_{1,13} ,a_{2,11} ,b_{2,11} ,a_{2,12} ,b_{2,12} ,a_{2,13} ,b_{2,13} } \right) \\ & \quad = \frac{1}{{4\pi^{2} }}\int_{0}^{2\pi } {\int_{0}^{2\pi } {R_{i}^{\left( 1 \right)} \cos \left| {2\tau_{2} - \tau_{1} } \right|} } {\text{d}}\tau_{1} {\text{d}}\tau_{2} . \\ \end{aligned}$$

Appendix D

In Sect. 3.3, the increment qi,2 is given by the following equation

$$\begin{aligned} q_{i,2} & = a_{i,21} \left( {\cos \tau_{1} - \cos 3\tau_{1} } \right) + b_{i,21} \left( {\cos \tau_{2} - \cos 3\tau_{2} } \right) \\ & \quad + {\kern 1pt} {\kern 1pt} a_{i,22} \left( {\cos \tau_{1} - \cos \left| {\tau_{2} - 2\tau_{1} } \right|} \right) + b_{i,22} \left( {\cos \tau_{2} - \cos \left( {\tau_{2} + 2\tau_{1} } \right)} \right) \\ & \quad + {\kern 1pt} a_{i,23} \left( {\cos \tau_{1} - \cos \left| {2\tau_{2} - \tau_{1} } \right|} \right) + b_{i,23} \left( {\cos \tau_{2} - \cos \left( {2\tau_{2} + \tau_{1} } \right)} \right) \\ & \quad + {\kern 1pt} a_{i,24} \left( {\cos \tau_{1} - \cos 5\tau_{1} } \right) + b_{i,24} \left( {\cos \tau_{2} - \cos 5\tau_{2} } \right) \\ & \quad + {\kern 1pt} a_{i,25} \left( {\cos \tau_{1} - \cos \left( {4\tau_{1} + \tau_{2} } \right)} \right) + b_{i,25} \left( {\cos \tau_{2} - \cos \left( {4\tau_{2} + \tau_{1} } \right)} \right) \\ & \quad + {\kern 1pt} a_{i,26} \left( {\cos \tau_{1} - \cos \left( {3\tau_{1} + 2\tau_{2} } \right)} \right) + b_{i,26} \left( {\cos \tau_{2} - \cos \left( {3\tau_{2} + 2\tau_{1} } \right)} \right) \\ & \quad + {\kern 1pt} a_{i,27} \left( {\cos \tau_{1} - \cos \left( {4\tau_{1} + \tau_{2} } \right)} \right) + b_{i,27} \left( {\cos \tau_{2} - \cos \left( {4\tau_{2} + \tau_{1} } \right)} \right) \\ & \quad + {\kern 1pt} {\kern 1pt} a_{i,28} \left( {\cos \tau_{1} - \cos \left| {4\tau_{1} - \tau_{2} } \right|} \right) + b_{i,28} \left( {\cos \tau_{2} - \cos \left| {4\tau_{2} - \tau_{1} } \right|} \right) \\ & \quad + {\kern 1pt} {\kern 1pt} a_{i,29} \left( {\cos \tau_{1} - \cos \left| {3\tau_{1} - 2\tau_{2} } \right|} \right) + b_{i,29} \left( {\cos \tau_{2} - \cos \left| {3\tau_{2} - 2\tau_{1} } \right|} \right). \\ \end{aligned}$$

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Zhao, D., Hao, P., Wang, J. et al. Surface effects on the quasi-periodical free vibration of the nanobeam: semi-analytical solution based on the residue harmonic balance method. Meccanica 55, 989–1005 (2020). https://doi.org/10.1007/s11012-020-01140-2

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