Jtam-A4.dvi


JOURNAL OF THEORETICAL SHORT RESEARCH COMMUNICATION

AND APPLIED MECHANICS

56, 3, pp. 887-891, Warsaw 2018
DOI: 10.15632/jtam-pl.56.3.887

STATIC RESONANCE IN ROTATING NANOBARS

Uǧur Güven

Yildiz Technical University, Department of Mechanical Engineering, Istanbul, Turkey

e-mail: uguven@yildiz.edu.tr

In this study, static resonance that occurs in rotating nanobars is addressed.The analysis is
based on Eringen’s nonlocal elasticity theory and is performed in Lagrangian coordinates.
Explicit solutions are given for both clamped-free and clamped-clamped boundary condi-
tions. The present study shows that the static resonance phenomenon is largely a critical
case requiring attention for rotating nanobars with small lengths.

Keywords: rotating nanobar, static resonance, nonlocal elasticity, lagrangian coordinates

1. Introduction

The rotating bars have attracted considerable attention inmechanical and aerospace engineering
applications asmachine elements such as turbines, propellers andhelicopter blades.As is known,
when the angular velocity of the bar reaches a certain critical value, the static resonance occurs
and the longitudinal displacement becomes unbounded.This phenomenon has been first noticed
byBhuta andJones (1963) and it hasbeen extendedbyBrunelle (1971) for the rotating disks.As
pointedout in (1963), theuse ofEulerian coordinates doesnot even showthis resonant character.
In those analyses (Bhuta and Jones, 1963; Brunelle, 1971) Lagrangian coordinates were used.
Shum and Entwistle (2006) reported that the linear uniaxial model is not representative for the
situation at larger strains due to higher angular velocity. The axial deformation of rotating rods
was investigated (Hodges andBless, 1994) by using two simpler nonlinear strain energymodels.
Nowadays, the recent developments in science and technology has enabled production of

various rotating structures in micro and nano scales. Some publications in this new field can
be found (Narendar, 2011, 2012; Narendar and Gopalakrishnan, 2011; Aranda et al., 2012;
Danesh andAsghari, 2014) in literature. However, no analytical or numerical study of the static
resonance in the nanobars has yet been done. The aim of this work is to investigate the scale
effect on the static resonance. In this analysis, the equation of motion is formulated in the
Lagrangian coordinates and Eringen’s nonlocal elasticity theory is adopted. In this study, the
static resonance phenomenon in rotating nanobars is addressed for two boundary conditions:
clamped-free (C-F) and clamped-clamped (C-C). It can be seen from the present analysis that
the results presented are strongly affected with the boundary conditions (C-F or C-C) and the
coordinates systems (Eulerian or Lagrangian).

2. Formulation of the problem and nonlocal elasticity solution

A uniform nanobar of lengthL rotating statically about the axis of rotation with angular velo-
city Ω is shown in Fig. 1.
The equation of motion in the Lagrangian coordinates is expressed (Bhuta and Jones, 1963)

as

∂σ

∂x
+ρΩ2(u+x)= 0 (2.1)



888 U. Güven

Fig. 1. Sketch of the rotating nanobar

where σ is the nonlocal longitudinal stress, ρ is density, x is the axial distance and u is the
longitudinal displacement.
Eringen’snonlocal elasticity theory (Eringen, 2002) for onedimensional case canbeexpressed

in the following form

σ− (e0a)
2∂
2σ

∂x2
=σlocal =E

∂u

∂x
(2.2)

where eoa is the small scale coefficient.
By using Eqs. (2.1) and (2.2), the nonlocal longitudinal stress is obtained as follows

σ=E
∂u

∂x
− (e0a)

2ρΩ2
(

1+
∂u

∂x

)

(2.3)

Substituting Eq. (2.3) into Eq. (2.1), the governing equation is given by

E
∂2u

∂x2
− (e0a)

2ρΩ2
∂2u

∂x2
+ρΩ2u=−ρΩ2x (2.4)

and the general solution to Eq. (2.4) becomes as

u=C1 sinkx+C2coskx−x (2.5)

whereC1 andC2 are integration constants and k
2 = ρΩ2/[E− (e0a)

2ρΩ2].
For the clamped-free boundary conditions, i.e. u(0) = 0 and u′(L) = 0, the longitudinal

displacement u is given by

u=
sinkx

kcoskL

[

1+
(e0a)

2ρΩ2

E− (e0a)
2ρΩ2

]

−x (2.6)

provided thatΩ does not correspond to a root of

coskL=0 (2.7)

WhenΩ corresponds to a root of Eq. (2.7)

Ωn =

√

√

√

√

√

√

√

E
ρ

[

(2n−1) π
2L

]2

1+(e0a)2
[

(2n−1) π
2L

]2
(2.8)

static resonances occur. The practical value of the critical angular velocity is obtained forn=1.



Short Research Communication – Static resonance in rotating nanobars 889

On the other hand, in the Eulerian coordinates, i.e. neglecting the longitudinal displacement
term u in Eq. (2.1), and by repeating the previous similar operations, the final form of the
longitudinal displacement is obtained as

u=
ρΩ2

2E

{

[L2+2(e0a)
2]x−

x2

3

}

(2.9)

Thus, Eq. (2.9) shows clearly that the static resonance phenomenon would not be noticed (i.e.,
the longitudinal displacement can not become unbounded for a certain value of the angular
velocity), for clamped-free boundary conditions when the Eulerian coordinates are used.
Secondly, for the clamped-clamped boundary conditions, i.e. u(0) = 0 and u(L) = 0, the

longitudinal displacement u is given by

u=
Lsinkx

sinkL
−x (2.10)

provided thatΩ does not correspond to a root of

sinkL=0 (2.11)

WhenΩ corresponds to a root of Eq. (2.11)

Ωn =

√

√

√

√

√

√

√

E
ρ

(

nπ
L

)2

1+
(

e0a2
nπ
L

)2
(2.12)

static resonances occurs. The critical angular velocity of practical interest is obtained forn=1.
If the Eulerian coordinates are used in the same analysis, longitudinal displacement expres-

sion (2.10) takes the following form

u=
ρΩ2

6E
(L2−x2)x (2.13)

Thus, Eq. (2.13) shows clearly that the static resonance phenomenon ofmotion can not be seen
for the clamped-clamped boundary conditions if the Eulerian coordinates are used and, further-
more, the longitudinal displacement is independent of the effect of the small scale coefficient.

3. Numerical example

In this Section, for a numerical example as in (Narendar and Gopalakrishnan, 2011), a(5,5)
SWCNT is considered. The diameter is d = 0.675nm, length L = 10d, the elasticity modulus
E =5.5TPa and density 2300kg/m3. In the numerical illustration the following defined ratio is
used:

Critical angular velocities ratio=Critical angular velocity calculated fromthenonlo-
cal elasticity theory/Critical angular velocity calculated from the classical elasticity
theory

Figure 2 shows the critical angular velocities ratiowith the dimensionless scale coefficient e0a/L,
for the clamped-free and the clamped-clampedboundary conditions. FromFig. 2 it is found that
as the scale coefficient e0a increases, thecritical angularvelocity decreases.Theclassical elasticity
solution overestimates the critical angular velocities compared to the nonlocal elasticity solution.
In addition, for the clamped-free boundary condition, the critical angular velocities are found
to be higher compared to those for the clamped-clamped boundary condition. For the range of
small scale parameters in Fig. 2, a detailed previous reference work (Narendar et al., 2011) has
been taken into consideration.



890 U. Güven

Fig. 2. Critical angular velocities ratio with dimensionless nonlocal scale coefficients

4. Conclusions

In this work, the static resonance phenomenon is investigated for rotating nanobars under
clamped-free and clamped-clamped boundary conditions. Here, the classical linear uniaxial mo-
del is extended by adopting Eringen’s nonlocal elasticity theory, and the equation of motion is
formulated in the Lagrangian coordinates. If the critical angular velocities obtained from the
nonlocal elasticity calculations are very small, as compared to those from the local elasticity
calculations, this linear uniaxial model can be reliably used, as indicated by Hodges and Bless
(1994) in detail. Hence, it should be noted that the linear uniaxial model used here will give
more reliable results with an increase in the scale coefficient for nanobars with sufficiently small
lengths under the clamped-clamped boundary conditions. The present analysis based on the
nonlocal elasticity theory shows that the static resonance can be a primary critical case for the
rotating nanobars having very small lengths, in contrast to the classical elasticity theory.

References

1. Aranda-Ruiz J.,LoyaJ.,Fernandez-SaezJ., 2012,Bendingvibrationsof rotatingnonuniform
nanocantilevers using the Eringen nonlocal elasticity theory,Composite Structures, 94, 2990-3001

2. BhutaP.G., JonesJ.P., 1963,Onaxialvibrationsof awhirlingbar,The Journal of theAcoustical
Society of America, 35, 217-221

3. Brunelle E.J., 1971, Stress redistribution and instability of rotating beams and disks, America
Institute Aeronautics Astronautics, 9, 758-759

4. Danesh V., Asghari M., 2014, Analysis of micro-rotating disk based on the strain gradient
elasticity,Acta Mechanica, 225, 1955-1965

5. Eringen A.C., 2002,Nonlocal Continuum Field Theories, Springer, NewYork

6. HodgesD.H.,BlessR.R., 1994,Axial instabilityof rotating rods revisited, International Journal
of Non-linear Mechanics, 29, 879-887

7. Narendar S., 2011,Mathematical modelling of rotating single-walled carbon nanotubes used in
nanoscale rotational actuators,Defence Science Journal, 61, 317-324



Short Research Communication – Static resonance in rotating nanobars 891

8. Narendar S., 2012, Differential quadrature based nonlocal flapwise bending vibration analysis of
rotating nanotube with consideration of transverse shear deformation and rotary inertia, Applied
Mathematics and Computation, 219, 1232-1243

9. Narendar S., Gopalakrishnan S., 2011, Nonlocal wave propagation in rotating nanotube,
Results in Physics, 1, 17-25

10. Narendar S., Roy Mahapatra D., Gopalakrishnan S., 2011, Prediction of nonlocal scaling
parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural
mechanics,nonlocal elasticityandwavepropagation, International Journal ofEngineering Sciences,
49, 509-522

11. ShumW.S., EntwistleR.D., 2006, Longitudinal vibration frequencies of steadilywhirling rods,
The Journal of the Acoustical Society of America, 119, 909-916

Manuscript received December 1, 2017; accepted for print January 10, 2018