Dynamics of noncontact rack-and-pinion device: Periodic back-and-forth motion of the rack
Mojtaba Nasiri,
1
Ali Moradian,
1
and MirFaez Miri
2,
*
1
Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45195-1159, Zanjan 45195, Iran
2
Department of Physics, University of Tehran, P.O. Box 14395-547, Tehran, Iran
Received 4 July 2010; published 1 September 2010
We study a nanoscale system composed of one corrugated cylinder pinion and one corrugated plate rack.
The pinion and rack have no mechanical contact, but are coupled via the lateral Casimir force. We consider the
case where the rack position versus time is a periodic triangular signal. We find that the device can rectify the
periodic but nonsinusoidal motion of the rack. Using the typical values of parameters, we find that the pinion
rotates with an average angular velocity =1 100 Hz. Experimental observation of the pinion rotation will
show that the quantum vacuum can intermesh the noncontact parts of nanomachines.
DOI: 10.1103/PhysRevE.82.037101 PACS numbers: 07.10.Cm, 85.85.j, 42.50.Lc, 46.55.d
I. INTRODUCTION
As a key interaction at nanoscale, the Casimir force 1,2
influences the dynamics of small devices. Chan and collabo-
rators have experimentally demonstrated frequency shifts,
hysteretic behavior, and bistability caused by the Casimir
force in the frequency response of a periodically driven mi-
cromachined torsional oscillator 3.
Two sinusoidally corrugated surfaces experience the lat-
eral Casimir force, as has been predicted 4,5 and verified
experimentally 6. Recently it has been suggested that the
lateral Casimir force may intermesh the noncontact parts of
nanomechanical devices 7–11. This gives a partial solution
to the wear problem in nanoscale mechanical systems 12.
Ashourvan, Miri, and Golestanian studied a rack and pin-
ion with no contact but coupled via the quantum vacuum.
The pinion is subject to an external load and experiences
friction when rotates around its axis. It is shown that both
uniform 8 and sinusoidal 9 motion of the rack can be
converted into uniform motion of the load. In this Brief Re-
port, we consider the case where the rack position versus
time is a periodic triangular signal. There are three reasons
for our study. First, from an experimental point of view, it
seems easier to enforce a rack to undergo a bidirectional
rather than a unidirectional motion. Realization of a high
velocity unidirectional motion requires a rack of great length.
Second, it is an immediate question whether the device can
rectify periodic but nonsinusoidal motion of the rack. Third,
we focus on a heavily damped system, so that inertia can be
neglected. We show explicitly that the rectified motion of the
pinion is a consequence of the inherent nonlinearity of the
system.
Here, we consider the noncontact rack and pinion device
shown schematically in Fig. 1a. Two harmonically corru-
gated plates with identical wavelength and lateral displace-
ment x - y experience a lateral Casimir force F
lateral
=
-F sin
2
x - y. The amplitude of the lateral Casimir force
depends on the distance H between the pinion and rack, cor-
rugation wavelength , corrugation amplitudes a
p
and a
r
,
and radius R 4,5,8. Recently, the lateral Casimir force in a
variety of complex geometries have gained much attention,
see e.g., 13–18. The Casimir torque plays the key role in
the equation of motion
- RF sin
2
x - y
-
R
dx
dt
- rW =0 1
for the coordinate x = R, where is the angle of rotation and
is the rotational friction coefficient. In the overdamped mo-
tion of the pinion, and / FR
2
are the natural units of
length and time, respectively. We define the scaled variables
X = x / , Y = y / , and T = FR
2
t / . We assume that the rack
position y versus time t is a periodic triangular signal as
shown in Fig. 1b. The periodic signal can be characterized
with the parameters y
0
, T
, T
1
, and S
1
. In its first period
*
miri@iasbs.ac.ir
λ
←→
R
r
W
y →
a
p
a
r
↑
|
V
p
→
x
H
FR
2
ζ λ
t T
1
T
∗
y
λ
y
0
λ
a)
b)
FIG. 1. Color onlinea The schematics of the rack and pinion
device. The pinion and rack have sinusoidal corrugations of wave-
length and amplitudes a
p
and a
r
, respectively. The rectified mo-
tion of the pinion manifests in a positive average velocity V
p
, while
working against an external load W. b The rack position versus
time is a periodic triangular signal, characterized by y
0
, T
, T
1
, and
S
1
.
PHYSICAL REVIEW E 82, 037101 2010
1539-3755/2010/823/0371014 ©2010 The American Physical Society 037101-1