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The dispersion characteristics of the waves propagating in a spinning single-walled carbon nanotube

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Abstract

As the nano-motor becomes a mechanical reality, its prototype can be envisaged as nano-sized rotating machinery at a situation, albeit for different purposes, like that in the first half of the 20th century during which rotor dynamics has contributed to boosting machine power capacity. Accordingly, we take the benefit of hindsight to develop a classical framework of vibration analysis. Essentially, the equations of motion are formulated to cope with both the special carbon-nanotube properties and the first author’s previously developed spinning beam formalism, establishing a model satisfactorily verified by some available molecular dynamics (MD) data and classical spinning beam results extracted from the literature. The model is inexpensive based on continuum mechanics as an alternative to the less-flexible MD method for simulating wave motion of the spinning single-walled carbon nanotube, yielding several interesting phenomena, including the fall-off and splitting of the wave characteristic curves and the unexpected gyroscopic phase property. Potential applications are proposed.

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Abbreviations

a :

CNT length scale

A :

cross-sectional area

B :

CNT eight-coefficient bearing matrix

c s :

shear wave velocity \(\sqrt {\kappa G/\rho } \)

c o :

longitudinal wave velocity \(\sqrt {E/\rho } \)

c sα :

phase speed of shear wave in nonlocal Timoshenko beam \(c_s /\sqrt {\alpha '} \)

c oα :

phase speed of longitudinal wave in nonlocal Timoshenko beam \(c_o /\sqrt {\alpha '} \)

C :

bearing damping coefficient matrix

C xx , C xy , C yx , C yy :

bearing damping matrix coefficients

C :

CNT rotor dynamic matrix

e o :

CNT constant

E :

Young’s modulus

eiµ :

phase angular velocity \((\dot \phi _{xo} ,\dot \phi _{yo} )\) relative to spin Ω

eiθ :

phase of spin Ω relative to angular velocity \((\dot \phi _{xo} ,\dot \phi _{yo} )\) of cross-section

F x (z 1,2, t), F x (z 1,2, t):

bearing forces in x and y respectively, at location z 1 or z 2

G :

shear modulus =E / 2(1 + υ)

h :

wall thickness

I :

second moment of area

J :

rotary inertia = ρI

k :

wavenumber (rad/nm)

K :

bearing stiffness coefficient matrix

K xx , K xy , K yx , K yy :

bearing stiffness matrix coefficients

L :

length of CNT rotor

m :

mass per unit length =ρA

m α :

effective mass at wavenumber k \(m\sqrt {\alpha '} = \rho A\sqrt {\alpha '} \)

q :

ratio of beam rotation to translation

r g :

radius of gyration = \(\sqrt {I/A} \)

r :

length scale of wave =c o r g /c s

t :

time (ps)

T l :

kinetic energy of original Timoshenko beam

T trans,nl :

nonlocal elasticity-induced translational kinetic energy of nonlocal Timoshenko beam

T rot,nl :

nonlocal elasticity-induced rotational kinetic energy of nonlocal Timoshenko beam

T TB-E :

total kinetic energy of nonlocal Timoshenko beam = T l T trans,nl + T rot,nl + T S

T S :

kinetic energy of beam spin

V :

interlayer surface potential, Tersoff-Brenner or Lennard-Jones

w x (z 1,2, t), w x (z 1,2, t), \(\dot w_x \)(z 1,2, t), \(\dot w_x \)(z 1,2, t):

journal displacements velocities at z 1 or z 2

w o :

arbitrary constant translation amplitude

w xo , w yo :

centroidal translation in the respective, x o , y o axes

\(\dot w_{xo} ,\dot w_{yo} \) :

centroidal translational velocities in the respective, x o , y o axes

x, y, z :

fixed (inertial) coordinates (nm)

x o , y o , z o :

spinning (floating) coordinates (nm)

z 1, z 2 :

bearing locations

α′:

nonlocal k-wave factor = 1+η 2 k 2

α :

nonlocal elasticity operator =1−η 2 ∂ 2 z

η :

Eringen’s nonlocal parameter =e o a

\(\dot \phi _{xo} \phi _{yo} ,\dot \phi _{yo} \phi _{xo} \) :

bending projection on spin axis

φ xo , φ yo :

cross-sectional bending rotation about the respective x o , y o axes

\(\dot \phi _{xo} ,\dot \phi _{yo} \) :

cross-sectional angular velocities of bending about the respective x o , y o axes

γ xo , γ yo :

shear deformations about the respective x o , y o axes

κ :

shear coefficient

λ :

wavelength (nm) =2π/k

Ω:

spinning velocity (rad/ps) and \(\Omega ' = \frac{\Omega } {{\omega _{cr} }} \)

Ωφ yo , Ωφ yo :

spin velocity projections in the cross section along respective x o and y o axes

ρ :

mass per unit volume (density)

υ :

Poisson’s ratio

ω :

wave frequency (rad/ps) and \(\omega ' = \frac{\omega } {{\omega _{cr} }} \)

ω cr :

critical frequency =c s /r g = c o /r

ω crα :

nonlocal critical frequency = \(\omega _{cr} /\sqrt {\alpha '} \)

ϖ xo :

angular velocity in x o

ϖ yo :

angular velocity in y o

ϖ zo :

angular velocity in z o axis

\(\square _{s\alpha }^2 \equiv \partial _z^2 - \frac{1} {{c_s^2 }}\alpha \partial _t^2 \) :

D’Alembertian for shear wave in nonlocal Timoshenko beam

\(\square _{o\alpha }^2 \equiv \partial _z^2 - \frac{1} {{c_o^2 }}\alpha \partial _t^2 \) :

D’Alembertian for longitudinal wave in nonlocal Timoshenko beam

∂ t ↔(·):

once time derivative

∂ 2 t ↔(··):

twice time derivative

∂ z , ∂ zo ,↔(′):

once space derivative

∂ 2 z , ∂ 2 zo ,↔(″):

twice space derivative

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Authors and Affiliations

  1. Mechanical Engineering Department, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

    K. T. Chan

  2. State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    YaPu Zhao

Authors
  1. K. T. Chan
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  2. YaPu Zhao
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Correspondence to YaPu Zhao.

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Recommended by Zhao YaPu (Editorial Board Member)

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Chan, K.T., Zhao, Y. The dispersion characteristics of the waves propagating in a spinning single-walled carbon nanotube. Sci. China Phys. Mech. Astron. 54, 1854 (2011). https://doi.org/10.1007/s11433-011-4476-9

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