1
Supplementary Information
Nonlinear
Mode
-
Coupling
in Nanomechanical Systems
M. H. Matheny
1
, L. G. Villanueva
1
, R. B. Karabalin
1
, J. E. Sader
1,2
, M. L. Roukes
1
1
Kavli Nanoscience
Institute and Departments of Physics, Applied Physics, and Bioengineering,
California Institute of Technology, Pasadena, California 91125
2
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010,
Australia
Derivation of the a
nalytical form of the nonlinear coupling coefficients
The flexural motion of a doubly clamped beam
generates a uniform strain along the beam
axis
,
because
the ends are clamped. This strain
gives rise to a uniform
tensile stress
via Hooke’s law,
which
increase
s
the beam stiffness.
This
can affect both the
frequency of the mode at which the
beam is excited (self
-
tuning) or those of other modes
(cross
-
tuning).
Here we derive the relevant
equations that describe these processes.
We start by solving for the
flexural
modes of a doubly clamped beam. T
he
deflection functions
of these modes
are given by
1
Φ
n
(
ξ
)
=
Q
n
(
cosh
휅
푛
휉
−
cos
휅
푛
휉
+
cosh
휅
푛
−
cos
휅
푛
sinh
휅
푛
−
sin
휅
푛
[
sin
휅
푛
휉
−
sinh
휅
푛
휉
]
)
,
S.1
where
휉
is
scaled
distance along the beam
normalized by the beam length
,
푄
푛
is the nth mode’s
amplitude normalization constant
(
푄
푛
≈
1
)
, and
휅
푛
=
√
휌퐴
푌퐼
휔
푛
2
4
.
The
Euler
-
Bernoulli equation
in the
presence of an axial
stress
is
2
2
휕
2
푢
2
휕
푡
2
+
훾
2
휕
4
푢
2
휕
ξ
4
−
훾
2
퐴
퐼
(
푇
푙
2
푌
+
1
2
∫
(
휕
푢
2
휕휉
)
2
1
0
푑휉
)
휕
2
푢
2
휕
휉
2
⏞
푎푥푖푎푙
푠푡푟푒푠푠
푐표푛푡푟푖푏푢푡푖표푛
=
0
,
훾
2
=
푌퐼
휇
푙
4
,
S.2
where
푢
2
is the beam
instantaneous
displacement
normal to the be
am axis
,
μ
is the linear mass
density of the beam,
Y
is the material Young’s modulus,
I
is the beam’s areal moment,
l
the length of
the beam,
A
the cross
-
sectional area, and
T
the intrinsic
axial tension
(stress)
of the material
. The
middle
expression
is t
he
contribution due to uniform axial
stress
in
the beam.
Within this
expression, t
he first term
gives
the
contribution arising
from intrinsic tension, and the second
accounts for
extension along the
beam
length
due to finite oscillation
amplitu
de
.
We then
decompose the
beam
motion into
its
normal modes
푢
2
(
휉
,
푡
)
=
∑
풜
푛
Φ
n
(
ξ
)
ζ
n
(
t
)
.
푛
S.3
S
ubstituting equation
S.3
into equation
S.2,
gives
∑
풜
푛
훷
푛
휁
푛
̈
+
훾
2
푛
∑
풜
푛
훷
푛
퐼푉
푛
휁
푛
−
훾
2
퐴
퐼
∑
풜
푛
훷
푛
′′
푛
휁
푛
(
푇
푙
2
푌
+
1
2
∫
(
∑
풜
푛
훷
푛
′
푛
휁
푛
)
2
1
0
푑휉
)
=
0
,
S.4
For a beam excited at two
of its modes
k
and
j
, we obtain
ζ
k
̈
+
ω
k
,
0
2
ζ
k
+
η
k
ω
k
,
0
2
ζ
k
[
푋
푘푘
푇
푙
2
푌
+
1
2
풜
푘
2
휁
푘
2
푋
푘푘
2
+
풜
푗
2
휁
푗
2
(
푋
푘푘
푋
푗푗
2
+
X
푗푘
2
)
]
=
0
,
S.5
where
휂
푘
=
퐴
퐼
1
∫
Φ
k
Φ
k
IV
푑휉
1
0
,
푋
nm
=
∫
Φ
푛
′
Φ
푚
′
푑휉
,
1
0
ω
k
,
0
2
=
∫
Φ
k
Φ
k
IV
푑휉
1
0
∗
훾
2
.
The summation convention is not used
.
The
resonant frequency
of each mode
is modified by the
intrinsic
tension,
T
, according
to
3
ω
k
,
t
2
=
ω
k
,
0
2
(
1
+
η
k
푋
푘푘
푇
푙
2
퐸
)
,
τ
k
=
η
k
1
+
η
k
푋
푘푘
푇
푙
2
퐸
.
S.6
Assuming the beam motion is weakly perturbed by the nonlinearity in Eq. (S.2), we then use
the
harmonic approximation
. Substituting
ζ
(
t
)
≅
cos
휔
푘
,
푡
푡
into Eq. (S.5)
gives the required resonant
frequency of mode
k
in the presence of finite oscillation amplitude of mode
s
k
and
j
,
휔
푘
,
푚표푑
2
=
휔
푘
,
푡
2
(
1
+
2
휆
푘푘
풜
푘
2
+
2
휆
푗푘
풜
푗
2
)
.
S.7
Thus, fro
m equation S.5 the change in
resonant
frequency
of
mode
k
is
훥
휔
푘
휔
푘
=
휆
푗푘
풜
푚푎푥
,
푗
2
,
S.
8
assuming that only mode
j
is driven to high amplitudes
, i.e., the tension developed by the oscillation
of mode k is insignificant
.
The coupling
coefficients
in
Eq. (S.9) are
휆
푗푘
=
(
2
−
훿
푗푘
)
휏
푘
8
(
푋
푘푘
푋
푗푗
2
+
푋
푗푘
2
)
.
S.
9
These coefficients
휆
푗푘
form the nonlinear stiffness
tensor that
relates
the
change in
resonant
frequency
to the
amplitude of resonant motion
of modes
k
and
j
. The diagonal components are
the
well
-
known “Duffing”
3
terms
of a single mode oscillation
. The off
-
diagonal components
describe the
nonlinear
coupling between two different modes.
4
REFERENCES
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Cleland, A. N.,
Foun
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-
state theory to device applications
.
Springer: Berlin ; New York, 2003; p xii, 436 p.
2.
Lifshitz, R.; Cross, M. C.,
Nonlinear Dynamics of Nanomechanical and Micromechanical
Resonators
. Wiley
-
VCH Verlag GmbH & Co. KGaA
: 2009; p 1
-
52.
3.
Postma, H. W. C.; Kozinsky, I.; Husain, A.; Roukes, M. L.,
Dynamic range of nanotube
-
and
nanowire
-
based electromechanical systems
. AIP: 2005; Vol. 86, p 223105.