of 12
PHYSICAL REVIEW B
108
, 224303 (2023)
Effect of intramodal and intermodal nonlinearities on the flexural resonant
frequencies of cantilevered beams
John E. Sader
,
1
,
*
Stefano Stassi
,
2
Carlo Ricciardi,
2
and Michael L. Roukes
3
1
Graduate Aerospace Laboratories and Department of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
2
Department of Applied Science and Technology, Politecnico di Torino,
Corso Duca Degli Abruzzi, 24, 10129 Torino, Italy
3
Departments of Physics, Applied Physics, and Bioengineering, California Institute of Technology,
Pasadena, California 91125, USA
(Received 13 July 2023; revised 24 October 2023; accepted 31 October 2023; published 5 December 2023)
Sensing applications that utilize nanomechanical resonators require careful control of nonlinear effects in
their eigenmodes to ensure robust measurement. While the effect of intra- and intermodal nonlinearities on the
resonant frequencies of doubly clamped elastic beams have been widely studied using theory and experiment,
commensurate studies on cantilevered beams are limited in comparison. Here, we present such a detailed study
that includes an explicit and simple formula for the flexural resonant frequencies of slender cantilevered beams
that accounts for intra- and intermodal nonlinearities. Using this general theory, numerical results for the modal
nonlinear coefficients are tabulated for the first 20 flexural eigenmodes of cantilevered beams possessing uniform
cross sections. The accuracy of this theory, and the effect of cantilever aspect ratio (length
/
width) on these
nonlinear coefficients, is explored using high-accuracy laser Doppler vibrometry experiments. We anticipate that
these results will find utility in single- and multimode applications, where the effect of finite oscillation amplitude
on the cantilever resonant frequencies can significantly impact measurement design and interpretation.
DOI:
10.1103/PhysRevB.108.224303
I. INTRODUCTION
Recent advances in the development of micro- and nano-
electromechanical systems (MEMS and NEMS, respectively)
have enabled a multitude of high-resolution sensing appli-
cations, ranging from the detection of air-borne analytes
to the imaging of surfaces with atomic resolution [
1
7
].
Miniaturization of mechanical resonators from macroscopic
to nanometer length scales dramatically enhances the mass
responsivity of the device due to an analyte adsorption
event—varying as the reciprocal of the fourth power of the
device size [
8
]. This has led to the development of highly
sensitive nanomechanics-based mass spectrometry capable of
detecting mass at the single nanoparticle and single molecule
level [
9
13
]. Multimode detection has also enabled other an-
alyte properties to be measured, including their stiffness and
shape [
14
18
].
Fundamental to all these applications is a comprehensive
understanding of the vibrational response of the resonator.
Driving the device at small oscillation amplitude leads to
operation in the quasilinear regime, facilitating interpreta-
tion of the measured resonant frequencies of a device. Even
so, a nonlinear response is inevitable at any oscillation am-
plitude, which can modify the device sensitivity and the
requisite theory for data analysis [
19
,
20
]. For example, multi-
modal detection of nanomechanical resonators has been used
to measure the mass distribution of an analyte [
16
,
21
] and
*
jsader@caltech.edu
characterize the nature of its interaction with the surrounding
fluid [
22
]. These applications required the effect of inter- and
intramodal nonlinearities to be negligible, which necessitated
careful tuning of the experiments to ensure a linear response
with no cross talk between the modes. Nonlinear effects thus
pose a fundamental limit on device operation in practice,
while simultaneously opening the door to the exploitation of
unprecedentedly rich dynamical features including the possi-
bility of enhanced measurement sensitivity [
23
]. Therefore, a
comprehensive understanding of these nonlinear effects, and
an ability to control them, is essential to enhancing mea-
surement sensitivity, device performance, and generating new
fields of application.
The Duffing resonator is a canonical model that incor-
porates a cubic nonlinearity into the usual simple harmonic
oscillator. A hardening nonlinearity in the Duffing model
accurately captures the resonance behavior of a slender dou-
bly clamped elastic beam, where increasing the oscillation
amplitude increases the resonant frequency of any single vi-
brational mode. This
intramodal
nonlinearity is driven by
induced tension along the beam axis, which is controlled by
the relative magnitude of the oscillation amplitude to the beam
thickness and can be easily calculated using Euler-Bernoulli
beam theory [
24
,
25
]. In contrast, a slender cantilevered elas-
tic beam cannot sustain such axial tension yet also exhibits
a weak hardening intramodal nonlinearity in its fundamen-
tal flexural mode, while higher-order flexural modes are
strongly softening [
26
28
]. This nonlinearity in cantilevered
beams is controlled by the beam length (not its thickness).
Thus much higher amplitudes are needed to induce nonlinear
2469-9950/2023/108(22)/224303(12)
224303-1
©2023 American Physical Society
SADER, STASSI, RICCIARDI, AND ROUKES
PHYSICAL REVIEW B
108
, 224303 (2023)
effects in cantilevered beams than those required for doubly
clamped beams. The mechanisms driving intramodal nonlin-
earities in doubly clamped and cantilevered beams can also
induce
intermodal
nonlinearities, where excitation of one vi-
brational mode affects the resonant frequencies of the other
modes. Intermodal nonlinearities of the flexural modes of
doubly clamped elastic beams have been studied both theoret-
ically and experimentally, where excellent agreement has been
reported [
29
34
]. In contrast, intramodal and particularly in-
termodal nonlinearities in cantilevered beams have received
comparatively little attention [
35
37
]. One notable study on
the effect of intermodal nonlinearities is that of Westra
et al.
[
35
] who primarily focused on the effect of flexural-torsional
intermodal nonlinearities in cantilevered beams. While some
comparison to experiment was reported, the study did not
involve accurate calibration of the oscillation amplitude nor
quantitative comparison between theory and measurement.
Moreover, the theory was not reported in a form that facilitates
implementation by practitioners.
We address this gap in the literature by reporting a compre-
hensive theory to describe the effect of intra- and intermodal
nonlinearities on the flexural resonant frequencies of slender
cantilevered beams of arbitrary cross section. This includes
(1) a simple yet general analytical formula for beams of ar-
bitrary cross section and (2) tabulated numerical nonlinear
coefficients for beams of uniform cross section that facilitate
implementation in practice. Specifically, in Sec.
II
we theo-
retically study the nonlinear response of slender cantilevered
beams. In contrast to doubly clamped beams, the leading-
order nonlinearity does not involve any induced tension but
arises from enhancement and competition between inertia
and stiffness of the beam at finite oscillation amplitude. This
theory is then used to derive a simple analytical formula
for the flexural resonant frequencies that accounts for intra-
and intermodal nonlinearities. Numerical data for the modal
nonlinear coefficients of a cantilevered beam of uniform cross
section are reported for the first 20 modes. Then, in Sec.
III
,
the developed theory and simple formula are experimentally
validated using highly accurate measurements. This involves
laser Doppler vibrometry (LDV) of microcantilever beams of
uniform cross section and varying aspect ratio (length
/
width),
facilitating rapid and accurate measurement of the oscillation
amplitude. Details of the theoretical derivation reported in
Sec.
II
are relegated to the Appendix.
II. THEORY
The theoretical framework in Ref. [
38
], for the nonlin-
ear dynamics of a slender cantilevered beam whose cross
section is uniform along its length, forms the basis for calcu-
lations reported in this study. This previous work is extended
here to (1) a beam of arbitrary cross section and variation
along the beam length in Sec.
II A
, which in turn is used to
(2) formulate the required general theory for the effect of
intra- and intermodal nonlinearities on the flexural resonant
frequencies in Sec.
II B
. These two items form the principle
theoretical contribution of this study, from which numerical
results for modal nonlinear coefficients are reported and then
compared to experiment.
FIG. 1. Schematic showing the undeflected (gray) and deflected
(red) slender cantilever beam of length,
L
, that is clamped at
x
=
0.
The inextensible condition of the cantilever beam is illustrated by the
fact that the total arc length of the beam is unchanged under deflec-
tion. The Cartesian coordinate system and its origin are indicated.
A. Action integral for nonlinear deflection
Consider a cantilevered beam of length,
L
, linear mass
density,
μ
, Young’s modulus,
E
, and areal moment of inertia,
I
. The action integral for the beam deflection function,
y
,
follows from the theoretical framework of Ref. [
38
], which for
a general beam with arbitrary cross-sectional variation along
its length becomes
S
(
y
)
=
L
2

t
1
t
0

1
0
μ

y
t

2
+
1
4
L
2

t

ζ
0

y
∂ζ


2
d
ζ


2
EI
L
4

2
y
∂ζ
2

2

1
+
1
L
2

y
∂ζ

2

d
ζ
dt
,
(1)
where
t
is time, the limits of integration,
t
0
and
t
1
, are arbitrary,
ζ
s
/
L
is the dimensionless arc length along the cantilever
beam from its clamped end, and
s
is the corresponding di-
mensional arc length. Only the leading-order nonlinear effect
of finite deflection,
y
, is included in Eq. (
1
). It is assumed that
the beam is inextensible, i.e., its axial strain is zero; see Fig.
1
.
The first term in Eq. (
1
) (proportional to
μ
) is the kinetic
energy density of the beam, whereas the second term is its
strain energy density. Equation (
1
) thus shows that both the
kinetic and strain energies are enhanced by finite deflection,
with the length scale for onset of these nonlinear effects being
the beam length,
L
; see the two terms with the prefactor 1
/
L
2
.
Nonlinear enhancement of the kinetic energy is due to the
motion of material points parallel to the undeflected beam
axis, i.e., in the
x
direction; see Fig.
1
. This effect is absent
in the (linear) small-deflection limit where the beam moves
in the (vertical)
y
direction only. In contrast, the increase in
strain energy is due to beam curvature at finite deflection, i.e.,
a curved beam is stiffer than a straight beam. These enhance-
ments in the kinetic and strain energies for finite (nonlinear)
deformation arise from the inextensible property of a deflected
cantilever beam.
224303-2
EFFECT OF INTRAMODAL AND INTERMODAL ...
PHYSICAL REVIEW B
108
, 224303 (2023)
B. Effect of intra- and intermodal nonlinearities
on the resonant frequencies
Consider a deflection function,
y
, of the form
y
(
ζ,
t
)
=

m
(
ζ
)
f
m
(
t
)
+

n
(
ζ
)
f
n
(
t
)
,
(2)
where

i
(
ζ
) are two eigenmodes (
i
=
m
,
n
) of the can-
tilevered beam in the linear small-amplitude limit, with
i
=
1
,
2
,
3
,...
, under the normalization

i
(
ζ
0
)
=
1,
ζ
0
is the
measurement position, and
f
i
(
t
) is the amplitude function of
eigenmode
i
. Substituting Eq. (
2
) into Eq. (
1
) and using the
orthogonal properties of the eigenmodes gives
S
(
y
)
=
L
2

t
1
t
0
i
=
m
,
n
I
(1)
i
̇
f
2
i
+
I
(2)
i
f
2
i
+
1
L
2
i
=
m
,
n
j
=
m
,
n
p
=
m
,
n
q
=
m
,
n
f
i
̇
f
j
f
p
̇
f
q
I
(3)
ijpq
f
i
f
j
f
p
f
q
I
(4)
ijpq
dt
,
(3)
where the time derivative
̇
f
i
(
t
)
df
i
(
t
)
/
dt
and
I
(1)
i
=

1
0
μ
2
i
d
ζ,
I
(2)
i
=

1
0
EI
L
4
(


i
)
2
d
ζ,
I
(3)
ijpq
=

1
0
μ


ζ
0


i


j
d
ξ

ζ
0


p


q
d
ξ


d
ζ,
I
(4)
ijpq
=

1
0
EI
L
4


i


j


p


q
d
ζ,
(4)
with the spatial derivative


i
(
ζ
)
d

i
(
ζ
)
/
d
ζ
.
We choose the time dependencies of eigenmodes,
m
and
n
,
to be
f
m
(
t
)
=
A
m
cos
(
ω
m
t
+
φ
m
)
,
f
n
(
t
)
=
A
n
cos
(
ω
n
t
+
φ
n
)
,
(5)
where the resonant frequencies,
ω
m
and
ω
n
, are to be evaluated
with
m

=
n
, the phases,
φ
m
and
φ
n
, are independent and arbi-
trary, and
A
m
and
A
n
are the amplitudes of eigenmodes
m
and
n
, respectively. Substituting Eq. (
5
) into Eq. (
4
) then gives the
required leading-order result for the relative frequency shift of
eigenmode,
m
, due to the intra- and intermodal nonlinearities,
m
ω
(0)
m
=
X
m

A
m
L

2
+
(
1
δ
mn
)
V
mn

A
n
L

2
,
(6)
where
i
ω
i
ω
(0)
i
, the linear resonant frequency of
eigenmode,
i
,is
ω
(0)
i
(
i
=
m
,
n
),
δ
mn
is the Kronecker delta
function, and the modal nonlinear coefficients are
X
m
=
3
I
(4)
mmmm
I
(2)
m
I
(3)
mmmm
4
I
(1)
m
,
(7a)
V
mn
=
I
(4)
mmnn
+
4
I
(4)
mnmn
+
I
(4)
nnmm
4
I
(2)
m

1
+
I
(2)
n
I
(1)
n
I
(1)
m
I
(2)
m

I
(3)
mnmn
4
I
(1)
m
.
(7b)
Note that the positive and negative terms in Eq. (
7
) arise from
different physical mechanisms. The (first) positive terms are
due to the nonlinear enhancement of beam stiffness and define
a hardening nonlinearity, whereas the (second) negative terms
arise from the increase in beam inertia and are softening.
These terms compete to define the overall nonlinear response
that can be either hardening or softening; as we show below,
intermodal nonlinearities always produce a softening effect,
i.e.,
V
mn
<
0.
The intermodal contribution in Eq. (
6
) is independent of
A
m
, i.e., the amplitude of eigenmode
m
. Equation (
6
) can
therefore be generalized to an arbitrary number of eigenmodes
by superposition, giving
m
ω
(0)
m
=
X
m

A
m
L

2
+
n
=
1
(
n

=
m
)
V
mn

A
n
L

2
,
(8)
where
m

1.
C. Cantilevered beam of uniform cross section
We apply the developed theory to a beam of uniform cross
section and a measurement position of
ζ
0
=
1, i.e., the free
end of the cantilevered beam. The required eigenmodes are

i
(
ζ
)
=
(
1)
i
2

cos(
C
i
ζ
)
cosh(
C
i
ζ
)
+
cos
C
i
+
cosh
C
i
sin
C
i
+
sinh
C
i
[cos(
C
i
ζ
)
cosh(
C
i
ζ
)]

,
(9)
with eigenvalues given by
cosh
C
i
cos
C
i
=−
1
,
(10)
for
i
=
1
,
2
,
3
,...
. The required coefficients for the intra- and
intermodal nonlinearities,
X
m
and
V
mn
, respectively, are then
obtained by substituting Eq. (
9
) into Eq. (
7
).
The intramodal nonlinear coefficient,
X
m
, grows rapidly
with
m
and has the high-mode-number asymptotic form,
X
m
∼−
π
4
m
4
48
,
m

1
,
(11)
which is derived using the corresponding asymptotic expres-
sion for the eigenmodes,

m
(
ζ
)
(
1)
m
+
1
2
sin

π

m
1
2

ζ
1
4

,
m

1
.
(12)
The boundary layers near the clamped and free ends,
ζ
=
0
and 1, respectively, are ignored. Equation (
11
) establishes
that the effect of the intramodal nonlinearities on the flexural
resonant frequencies increases strongly with mode number,
m
.
Combining Eq. (
11
) with accurate numerical solutions for
X
m
, using the true eigenmodes in Eq. (
9
), allows a semiempir-
ical expression to be formulated,
X
m
≈−
0
.
574456
+
3
.
85554
m
7
.
22798
m
2
+
5
.
99564
m
3
2
.
02936
m
4
,
(13)
that exhibits a maximum error of
<
3% for all
m
; see Fig.
2
.
This simple formula facilitates application in practice.
224303-3
SADER, STASSI, RICCIARDI, AND ROUKES
PHYSICAL REVIEW B
108
, 224303 (2023)
FIG. 2. Relative error of the approximate formula for the in-
tramodal nonlinear coefficient, Eq. (
13
), as a function of mode
number,
m
.
Similarly, we find the following asymptotic solution for the
intermodal nonlinear coefficient:
V
mn
∼−
π
2
16

n
m

2
×
m
6
2
m
5
n
+
3
m
4
n
2
+
3
m
2
n
4
2
mn
5
+
n
6
(
m
2
n
2
)
2
,
m
,
n

1
,
(14)
which is strictly negative. The quadratic term, (
n
/
m
)
2
,onthe
right-hand side of Eq. (
14
) is not symmetric in
m
and
n
,
while the remaining term is symmetric. This shows that in
the large-mode-number limit, the magnitude of
V
mn
generally
increases with mode number of the excited eigenmode,
n
,
and decreases with the mode number of the detected eigen-
mode,
m
. Moreover, the intermodal nonlinear coefficient is
enhanced by choosing
m
n
. These general observations can
be utilized in practice to control the effect of intermodal
nonlinearities. In contrast to the asymptotic solution for the
intramodal coefficient in Eq. (
11
), the complexity of Eq. (
14
)
hinders the development of a simple approximate expression
that is accurate for all
m
,
n
.
Numerical results for the intramodal nonlinear coefficients,
X
m
, and the corresponding intermodal coefficients,
V
mn
,are
given in Tables
I
and
II
, respectively, for the first 20 flexural
modes. The reported data sets are normalized by the cor-
responding asymptotic expressions for large mode number.
These results confirm the validity of the large-mode-number
asymptotic solutions in Eqs. (
11
) and (
14
) and show that non-
linear effects in slender cantilevered beams of uniform cross
section are predominantly softening. The single exception
is for the intramodal nonlinearity of the fundamental mode,
m
=
1. For this special case, contributions from nonlinear
inertia and nonlinear stiffness are finely balanced, with stiff-
ness dominating and producing a weak hardening response.
Generally, the effects of nonlinear inertia are enhanced with
TABLE I. Scaled intramodal nonlinear coefficient,
̄
X
m
X
m
/
X
asym
m
, for a cantilevered beam of uniform cross section, where
X
asym
m
is the large-
m
asymptotic solution in Eq. (
11
).
m
̄
X
m
m
̄
X
m
1
0.00955158
11
0.759420
2
0.198783
12
0.777430
3
0.340546
13
0.792946
4
0.454406
14
0.806448
5
0.536316
15
0.818304
6
0.597681
16
0.828797
7
0.645052
17
0.838147
8
0.682615
18
0.846531
9
0.713083
19
0.854092
10
0.738267
20
0.860944
mode number,
m
, leading to strong intramodal softening for
m

2. The situation is more complicated for the intermodal
nonlinearities, as is evident from Eq. (
14
), the data set in
Table
II
, and the discussion above.
III. EXPERIMENTAL VERIFICATION
Next, we report an experimental protocol to accurately
measure the nonlinear response of a nanomechanical device
and thereby assess the validity of the reported theoretical
model.
This assessment is performed on cantilevers of varying
slenderness, i.e., aspect ratio, AR
=
length
/
width. A com-
mercial atomic force microscope (AFM) chip is used, which
contains three tipless single-crystal-silicon cantilevers that
have been precisely micromachined to a rectangular paral-
lelepiped geometry; see Fig.
3
. All these cantilevers have
identically constant widths of 29 μm and thicknesses of 2 μm
and thus present a uniform cross section as per the theoretical
results and data set reported in Sec.
II C
. These cantilevers are
denoted by their lengths of 396 μm (long), 193 μm (medium),
and 94 μm (short). Therefore, their aspect ratios are AR
=
14 (long), 6.6 (medium), and 3.2 (short). The cantilever
FIG. 3. Scanning electron micrographs of the three microcan-
tilevers used in Sec.
III
, with the measured length and width of each
cantilever indicated; all microcantilevers are of width, 29 μm, and
have a nominal thickness of 2 μm.
224303-4
EFFECT OF INTRAMODAL AND INTERMODAL ...
PHYSICAL REVIEW B
108
, 224303 (2023)
TABLE II. Scaled intermodal nonlinear coefficient,
V
mn
/
V
asymp
mn
, for a cantilevered beam of uniform cross section, where
V
asymp
mn
is the large mode number asymptotic solution in Eq. (
14
).
n
1 2 3 4567891011121314151617181920
1
NA
0.181536 0.913752 0.697795 0.548215 0.408172 0.318249 0.250220 0.202805 0.166445 0.139415 0.118038 0.101387 0.0878430 0.0769216 0.0678303 0
.0603029 0.0539166 0.0485180 0.0438655
2 0.0739568
NA
0.0851124 0.954959 0.917832 1.04209 0.879624 0.795278 0.665388 0.580479 0.493607 0.431392 0.373582 0.329717 0.290140 0.258855 0.2
30851 0.208028 0.187595 0.170543
3 0.240372 0.0549583
NA
0.195532 0.923310 0.860550 1.20995 1.08995 1.12706 0.988439 0.931499 0.816328 0.747909 0.660776 0.601485 0.536751 0.4892
39 0.440812 0.403451 0.366676
4 0.151079 0.507510 0.160930
NA 0.304233 0.891432 0.789944 1.20407 1.11396 1.27013 1.15315 1.16505 1.04869 1.00713 0.907108 0.853024 0.771956 0.7
20181 0.655718 0.610550
5 0.106043 0.435792 0.678929 0.271808 NA 0.391209 0.878217 0.753285 1.15936 1.07697 1.30393 1.20885 1.29058 1.18627 1.18844 1.08877 1.05642 0.968
676 0.925051 0.850850
6 0.0733668 0.459774 0.588000 0.740065 0.363524 NA 0.460396 0.874485 0.738841 1.11440 1.03034 1.28789 1.20665 1.34093 1.25106 1.29839 1.20630 1.2
0414 1.11720 1.09089
7 0.0543260 0.368572 0.785148 0.622819 0.775016 0.437236 NA 0.516116 0.875440 0.736508 1.07785 0.990070 1.25429 1.17993 1.34728 1.26922 1.35454 1
.27220 1.30142 1.21964
8 0.0411091 0.320716 0.680719 0.913676 0.639800 0.799306 0.496734 NA 0.561730 0.878644 0.740449 1.04986 0.958972 1.21756 1.14593 1.33098 1.26106
1.37453 1.30128 1.35857
9 0.0323501 0.260529 0.683419 0.820713 0.956057 0.655678 0.818054 0.545390 NA 0.599662 0.882865 0.747520 1.02879 0.936026 1.18336 1.11244 1.3046
5 1.23983 1.37222 1.30628
m 10 0.0259346 0.222013 0.585467 0.914073 0.867521 0.966033 0.672271 0.833304 0.585757 NA 0.631654 0.887469 0.756036 1.01294 0.919478 1.15345 1.0
8245 1.27518 1.21333 1.35722
11 0.0213121 0.185217 0.541306 0.814192 1.03047 0.876276 0.965235 0.688959 0.846084 0.619708 NA 0.658975 0.892128 0.765082 1.00098 0.907734 1.12
799 1.05675 1.24604 1.18584
12 0.0177607 0.159328 0.466923 0.809667 0.940319 1.07810 0.872691 0.961506 0.705119 0.857001 0.648618 NA 0.682565 0.896673 0.774164 0.991906 0.8
99540 1.10656 1.03520 1.21884
13 0.0150527 0.136146 0.422113 0.719128 0.990578 0.996694 1.09092 0.866611 0.957559 0.720394 0.866457 0.673509 NA 0.703129 0.901019 0.783019 0.9
84984 0.893961 1.08860 1.01734
14 0.0128940 0.118797 0.368704 0.682793 0.900184 1.09504 1.01460 1.08781 0.861332 0.954238 0.734635 0.874737 0.695151
NA
0.721211 0.905130 0.791
508 0.979683 0.890310 1.07353
15 0.0111800 0.103511 0.332326 0.608946 0.892978 1.01162 1.14713 1.01376 1.07824 0.857687 0.951713 0.747812 0.882051 0.714132
NA
0.737230 0.9089
94 0.799571 0.975617 0.888085
16 0.00977415 0.0915580 0.294017 0.567729 0.811071 1.04089 1.07140 1.16737 1.00493 1.06671 0.855654 0.949927 0.759962 0.888560 0.730909
NA
0.751
518 0.912614 0.807185 0.972497
17 0.00862383 0.0810361 0.265967 0.509893 0.781027 0.959755 1.13478 1.09769 1.16966 0.993487 1.05524 0.854962 0.948758 0.771149 0.894392 0.7458
41
NA
0.764340 0.916001 0.814353
18 0.00765883 0.0725347 0.238033 0.472505 0.711355 0.951610 1.05866 1.18844 1.10409 1.16253 0.981967 1.04467 0.855309 0.948083 0.781451 0.89964
7 0.759214
NA
0.775909 0.919167
19 0.00685066 0.0650183 0.216553 0.427633 0.675248 0.877614 1.07648 1.11837 1.21467 1.09951 1.15092 0.971449 1.03529 0.856430 0.947794 0.790947
0.904406 0.771259
NA
0.786399
20 0.00616037 0.0587898 0.195754 0.396032 0.617739 0.852333 1.00340 1.16131 1.15007 1.22328 1.08941 1.13762 0.962307 1.02711 0.858110 0.947800 0
.799713 0.908736 0.782163
NA
224303-5
SADER, STASSI, RICCIARDI, AND ROUKES
PHYSICAL REVIEW B
108
, 224303 (2023)
TABLE III. Comparison of the measured intramodal nonlinear coefficients,
X
expt
m
, of the three cantilevers, to the (infinite aspect ratio)
theoretical model of Sec.
II C
,
X
theory
m
. Results are given to three significant figures for the first three flexural modes, except for the smallest AR
due to overlap of the flexural and torsional resonances. Listed uncertainties represent a 95% confidence interval arising from least-squares fits
to the measured amplitude responses; see Fig.
4
.
X
expt
m
Mode,
mX
theory
m
AR
=
3.2 (short)
AR
=
6.6 (medium)
AR
=
14 (long)
1
0.0194
0
.
0143
±
0
.
0017
0
.
0170
±
0
.
0031
0
.
0170
±
0
.
0031
2
6.45
12.1
±
2
.
2
7.03
±
1
.
01
6.27
±
0
.
19
3
56.0
64.9
±
3
.
1
56.9
±
1
.
41
dimensions are measured using a Zeiss MERLIN field emis-
sion scanning electron microscope, the results of which are
reported in Fig.
3
.
The AFM cantilever chip is mounted with carbon adhesive
tape on a piezoelectric disk that is used for actuation. This
system is contained in a vacuum chamber. All the measure-
ments are performed at room temperature and an air pressure
of 2
×
10
7
mbar, which is achieved using a vacuum system
comprising a membrane and turbomolecular pump (HiCube80
Eco, Pfeiffer). The vibrational amplitude response of the can-
tilevers is measured with a laser Doppler vibrometer (LDV,
MSA-500, Polytec Gmbh) coupled with a lock-in amplifier
(HF2LI, Zurich Instruments). The lock-in amplifier is used
both for piezodisk actuation and for LDV signal analysis, with
the procedure described below. Specifically, the lock-in ampli-
fier sweeps the frequency around the investigated resonance
mode and records the output of the LDV, which is intrinsically
amplitude calibrated and linear within the amplitude range in-
vestigated. The drive frequency is initially swept upwards and
downwards to identify whether the nonlinearity of the mode
in question is hardening or softening in nature. The frequency
sweep direction is then chosen to be upwards
/
downwards for
an initially observed hardening
/
softening nonlinearity. This
enables unambiguous measurement of the resonant frequency,
which coincides with the frequency of maximal oscillation
amplitude.
A. Intramodal nonlinearity
Simultaneously measuring the resonant frequency,
ω
m
,of
each eigenmode of the cantilever and the oscillation amplitude
at this resonance,
A
m
, allows the intramodal nonlinear coeffi-
cient,
X
expt
m
, to be experimentally determined using
m
ω
(0)
m
=
X
expt
m

A
m
L

2
,
(15)
where
m
ω
m
ω
(0)
m
. A sample measurement for the first
flexural mode,
m
=
1, of the short cantilever is given in Fig.
4
.
Using the same approach, intramodal nonlinear coefficients
are measured for the first three flexural eigenmodes of the
medium and long cantilevers. The third flexural eigenmode
of the short cantilever cannot be measured robustly because
this flexural mode and the first torsional mode have similar
resonant frequencies.
The measured intramodal nonlinear coefficients,
X
expt
m
,are
reported in Table
III
along with a comparison to the (infinite
aspect ratio) theory of Sec.
II C
. Good agreement is evident
between theory and measurement, especially at the largest
aspect ratio of AR
=
14. Reducing the cantilever aspect ratio
and
/
or increasing its mode number (which reduces its effec-
tive aspect ratio) makes the intramodal nonlinear coefficients
more negative, i.e., more softening. This is true for all mea-
sured eigenmodes,
m
=
1
,
2
,
3.
Interestingly, the degree of softening in measurements of
the fundamental flexural mode (for cantilevers of uniform
cross section) with decreasing aspect ratio is less than that
reported in Ref. [
27
] (for cantilevers of nonuniform cross
section). In the present measurements, this intramodal non-
linearity is always a hardening phenomenon, i.e,
X
expt
1
>
0,
whereas in Ref. [
27
] it becomes softening, i.e.,
X
expt
1
<
0,
for AR

8. Combining these independent measurements on
different cantilever types (uniform and nonuniform cross sec-
tions) leads to the conclusion that the hardening
/
softening
nature of the fundamental flexural mode can be controlled by
0
10
20
30
-0.5
0
0.5
1
measured data
maxima
linear fit to maxima
FIG. 4. Measured intramodal nonlinear coefficient,
X
expt
1
,ofthe
short cantilever, i.e., for eigenmode,
m
=
1. Solid (blue) curves are
measured data of the squared amplitude response of the free end of
the cantilever, as the drive amplitude is increased (bottom to top).
The solid (green) dots denote the maxima of the amplitude responses
(where the resonant frequencies are measured) and the straight (red)
dashed line is a linear fit to these maxima; the latter is used to
determine
X
expt
1
from Eq. (
15
).
224303-6
EFFECT OF INTRAMODAL AND INTERMODAL ...
PHYSICAL REVIEW B
108
, 224303 (2023)
FIG. 5. Experimental setup to measure the intermodal nonlinear
coefficients. Oscillation amplitude of a cantilever is measured using
LDV and acquired by a lock-in amplifier. The lock-in amplifier is
used to sweep the drive frequency of the excited eigenmode,
n
,and
measure the resonant frequency of a different detected eigenmode,
m
, with a digital PLL. Mechanical excitation of the cantilever is
performed using a piezoelectric shaker which is controlled by lock-in
outputs.
varying the cantilever cross section along its length. This is
not unexpected given the fine balance between the effects of
nonlinear stiffness (hardening) and nonlinear inertia (soft-
ening) in the fundamental flexural mode; see discussion in
Sec.
II C
.
B. Intermodal nonlinearity
To measure the intermodal nonlinear coefficients, we em-
ploy two input and two output channels of the lock-in
amplifier; see Fig.
5
. This configuration is used to detect
the frequency shift of one eigenmode that is induced by
the excitation of another eigenmode. Specifically, one set of
input
/
output channels is engaged to excite and sweep the
frequency of eigenmode,
n
, for which its oscillation amplitude
is measured using the LDV. The second set of input
/
output
channels runs a digital phase locked loop (PLL) that excites
and detects the resonant frequency of another eigenmode,
m
.
Note that the excited and detected eigenmodes differ, i.e.,
m

=
n
.
The detected eigenmode,
m
, is driven at low amplitude so
that it remains in its quasilinear regime, i.e., its resonant fre-
quency is independent of its oscillation amplitude. This avoids
any significant effect from its intramodal nonlinearity. In con-
trast, the excited eigenmode,
n
, is driven at relatively large
amplitude to generate a large frequency shift and facilitate
measurement. The required intermodal nonlinear coefficient,
V
expt
mn
, is then determined using
m
ω
(0)
m
=
V
expt
mn

A
n
L

2
,
m

=
n
,
(16)
by overlaying (1) the fractional frequency shift of eigenmode,
m
, with (2) the oscillation amplitude squared of eigenmode,
m
, both as functions of the drive frequency of eigenmode,
TABLE IV. Comparison of the measured intermodal nonlinear
coefficients,
V
expt
mn
, to the (infinite aspect ratio) theoretical model
of Sec.
II C
(values in parentheses). Results are reported to three
significant figures. The indices,
m
and
n
, correspond to the detected
and excited eigenmodes, respectively. Listed uncertainties represent
a 95% confidence interval arising from the least-squares fits.
n
m
123
1NA
2
.
77
±
0
.
04
41
.
6
±
0
.
6
(
2
.
84)
(
40
.
3)
2
0
.
0702
±
0
.
0009
NA
5
.
07
±
0
.
24
AR = 14
(
0
.
0722)
(
4
.
88)
3
0
.
124
±
0
.
002
0
.
660
±
0
.
007
NA
(
0
.
131)
(
0
.
623)
1NA
3
.
10
±
0
.
06
53
.
8
±
0
.
6
(
2
.
84)
(
40
.
3)
2
0
.
0718
±
0
.
0048
NA
6
.
63
±
0
.
17
AR = 6.6
(
0
.
0722)
(
4
.
88)
3
0
.
113
±
0
.
013
0
.
691
±
0
.
004
NA
(
0
.
131)
(
0
.
623)
1NA
4
.
00
±
0
.
24
(
2
.
84)
2
0
.
0751
±
0
.
0022
NA
AR = 3.2
(
0
.
0722)
3NA
n
. The single adjustable parameter,
V
expt
mn
,inEq.(
16
)isused
to achieve this overlay using a least-squares fit. A sample
measurement from this protocol is given in Fig.
6
, showing
a precise overlap in accordance with Eq. (
16
). To ensure a
linear response in eigenmode,
m
, this measurement protocol
is repeated for four small but different drive amplitudes of
eigenmode,
m
.
Intermodal coefficients,
V
expt
mn
, are measured for all permu-
tations, (
m
,
n
), of the first three flexural eigenmodes of the
long and medium cantilevers. For the short cantilever,
V
expt
12
and
V
expt
21
only are measured—this is for the same reason
the intramodal nonlinear coefficient of eigenmode 3 is not
measured; see Sec.
III A
. Table
IV
gives the measurement
data set and provides a comparison with the (infinite aspect
ratio) theoretical predictions in Table
II
. Good agreement is
observed throughout, with discrepancy increasing as the as-
pect ratio is reduced or the eigenmode numbers increased.
This is similar to observations for the intramodal nonlinearity
reported in Sec.
III A
and is consistent with the overriding
large-aspect-ratio assumption of the theoretical model.
IV. CONCLUSIONS
Nonlinearities in the eigenmodes of resonant mechanical
sensors intrinsically control the dynamic range over which
quasilinear measurements can be performed. Knowledge of
this phenomenon is therefore critical in experimental de-
sign and operation. We have reported a detailed theoretical
and experimental investigation of the modal nonlinearities
in cantilevered beams. Analytical formulas for the effect of
intra- and intermodal nonlinearities, on the flexural resonant
224303-7
SADER, STASSI, RICCIARDI, AND ROUKES
PHYSICAL REVIEW B
108
, 224303 (2023)
112
112.01 112.02 112.03 112.04 112.05 112.06
f
1
(kHz)
112
112.01 112.02 112.03 112.04 112.05 112.06
f
1
(kHz)
0
-2
-4
-6
-8
(
μ
m
2
)
A
2
2
0
1
2
3
4
0
-2
-4
-6
-8
0
-2
-4
-6
2
25 mV
35 mV
42.5 mV
50 mV
amplitude
frequency shift
error
rescaled-amplitude overlay
(c)
(a)
(d)
(b)
(×10
−5
)
+ 2.77
(
A
2
L
(
2
error =
Δ
f
2
f
2
(0)
(×10
−5
)
Δ
f
2
f
2
(0)
(×10
−5
)
Δ
f
2
f
2
(0)
FIG. 6. Measurement of the intermodal nonlinear coefficient,
V
expt
21
, for the long cantilever, i.e., the detected eigenmode is
m
=
2 and the
excited eigenmode is
n
=
1;
ω
i
=
2
π
f
i
. (a) Squared oscillation amplitude curves of the detected eigenmode,
m
=
2, and (b) its fractional
frequency shift, measured by sweeping the drive frequency around the resonant frequency of the excited eigenmode,
n
=
1. Measurements
taken for different drive voltages applied to the piezoshaker, as indicated in (a). (c) The squared oscillation amplitude curves in (a) are rescaled
(dashed yellow curves) to overlap with the frequency shift curves in (b). This gives the intermodal nonlinear coefficient
V
expt
21
=−
2
.
77
±
0
.
04
via Eq. (
16
). (d) Residual error of the fractional frequency shift and rescaled amplitude curves in (c); spikes correspond to slight mismatches
between the vertical jumps in (c).
frequencies of a slender cantilever beam of arbitrary cross
section, were derived. This theory was applied to a cantilever
of uniform cross section, for which (1) simple asymptotic
formulas in the high-mode-number limit and (2) accurate
tabulated numerical data for the first 20 eigenmodes were pre-
sented. Both intra- and intermodal nonlinearities were found
to induce a softening effect, with the singular exception of
the intramodal nonlinearity of the fundamental flexural mode
which is weakly hardening.
An experimental protocol was developed to accurately
measure the oscillation amplitudes and resonant frequencies
of cantilevered beams. This allowed for a rigorous assessment
of the developed theory. Specifically, this protocol was ap-
plied to three cantilevers of varying aspect ratio, for which
good agreement with theory was observed, especially for can-
tilevers of large aspect ratio. Decreasing the aspect ratio was
found to enhance the softening nonlinearity in all cases. A
theory for this aspect ratio phenomenon is yet to be reported
and represents an area of future work.
This study shows that elastic beam theory can be used with
confidence to calculate the effect of modal nonlinearities in
slender cantilevered beams with finite aspect ratio, which has
immediate implications to the development of highly sensitive
nanoelectromechanical sensors.
ACKNOWLEDGMENTS
The authors thank M. Matheny for useful discussions.
J.E.S. and M.L.R. acknowledge support from the
Kavli Nanoscience Institute at Caltech. This research
was partly supported by the Ministero dell’Istruzione,
dell’Università e della ricerca (MIUR), through the
PRIN2017–Prot.20172TZHYX grant.
224303-8
EFFECT OF INTRAMODAL AND INTERMODAL ...
PHYSICAL REVIEW B
108
, 224303 (2023)
APPENDIX: DERIVATION OF THE MODAL NONLINEARITY THEORY
In this Appendix, we present derivation details of (1) the leading-order nonlinear response of a slender cantilevered beam
and (2) formulas for the effect of modal nonlinearities on the flexural resonant frequencies, reported in Sec.
II
. This utilizes the
theoretical framework of Ref. [
38
]. It is assumed that the beam is inextensible, i.e., its axial strain is zero.
1. Action integral
The kinetic energy of the beam is
T
=
L
2

1
0
μ




u
(
ζ,
t
)
t




2
d
ζ,
(A1)
where
u
(
ζ,
t
)
=
X
(
ζ,
t
)
i
+
Y
(
ζ,
t
)
j
,
(A2)
is the local displacement vector of any material point along the beam with respect to its initial (undeformed) position,
i
and
j
are
the Cartesian basis vectors in the
x
and
y
directions, respectively, and
X
and
Y
are the corresponding Lagrangian displacement
variables; see Fig.
1
. The strain energy is
V
=
L
2

1
0
EI
κ
2
(
ζ,
t
)
d
ζ,
(A3)
where
κ
is the local curvature of the beam’s neutral axis. All symbols are as defined in Sec.
II
.
The shape of the deformed beam is parametrized by the dimensionless arc length,
ζ
,togive
r
=
x
(
ζ,
t
)
i
+
y
(
ζ,
t
)
j
,
(A4)
where (
x
,
y
) are the Cartesian coordinates of the (fixed) inertial frame, as per Fig.
1
. This is directly related to the Lagrangian
displacement variables in Eq. (
A2
)via
x
(
ζ,
t
)
=
ζ
L
+
X
(
ζ,
t
)
,
y
(
ζ,
t
)
=
Y
(
ζ,
t
)
.
(A5)
The general expression for the curvature is
κ
=
x
∂ζ
2
y
∂ζ
2
y
∂ζ
2
x
∂ζ
2


x
∂ζ

2
+

y
∂ζ

2

3
2
.
(A6)
However, because
ζ
is the dimensionless arc length, it follows that

x
∂ζ

2
+

y
∂ζ

2
=
L
2
(A7)
and Eq. (
A6
) simplifies,
κ
=
1
L
3

x
∂ζ
2
y
∂ζ
2
y
∂ζ
2
x
∂ζ
2

.
(A8)
Rearranging Eq. (
A7
) produces the following leading-order asymptotic expressions in the limit of small deflection:
x
∂ζ
=
L

1
1
2
L
2

y
∂ζ

2
+···

,
2
x
∂ζ
2
=−
1
L
2
y
∂ζ
2
y
∂ζ
+···
.
(A9)
Substituting Eq. (
A9
) into Eq. (
A8
) gives the required leading-order expression for the curvature,
κ
=
1
L
2
2
y
∂ζ
2

1
+
1
2
L
2

y
∂ζ

2
+···

.
(A10)
Similarly, substituting Eq. (
A5
) into Eq. (
A7
)gives

L
+
X
∂ζ

2
+

y
∂ζ

2
=
L
2
,
(A11)
224303-9
SADER, STASSI, RICCIARDI, AND ROUKES
PHYSICAL REVIEW B
108
, 224303 (2023)
which upon solution for
X
(again in the asymptotic limit of small
y
/∂ζ
) while imposing the clamp condition,
X
(0
,
t
)
=
0,
produces the leading-order asymptotic result,
X
(
ζ,
t
)
=−
1
2
L

ζ
0

y
∂ζ


2
d
ζ

+···
.
(A12)
The required leading-order expressions for the kinetic energy and strain energy of the beam are then obtained by substituting
Eqs. (
A5
) and (
A12
) into Eq. (
A1
), and Eq. (
A10
) into Eq. (
A3
), respectively,
T
=
L
2

1
0
μ

y
t

2
+
1
4
L
2

t

ζ
0

y
∂ζ


2
d
ζ


2
d
ζ,
(A13a)
V
=
1
2
L
3

1
0
EI

2
y
∂ζ
2

2

1
+
1
L
2

y
∂ζ

2

d
ζ.
(A13b)
The action,
S
(
y
)

t
1
t
0
T
Vdt
, where the limits of integration,
t
0
and
t
1
, are arbitrary, then follows from Eqs. (
A13
),
S
(
y
)
=
L
2

t
1
t
0

1
0
μ

y
t

2
+
1
4
L
2

t

ζ
0

y
∂ζ


2
d
ζ


2
EI
L
4

2
y
∂ζ
2

2

1
+
1
L
2

y
∂ζ

2

d
ζ
dt
.
(A14)
2. Effect of modal nonlinearities on the flexural resonant frequencies
We consider a deflection function,
y
(
ζ,
t
)
=

m
(
ζ
)
f
m
(
t
)
+

n
(
ζ
)
f
n
(
t
)
,
(A15)
where all symbols in this section are as defined in Sec.
II
. Substituting Eq. (
A15
) into Eq. (
A14
)gives
S
(
y
)
=
L
2

t
1
t
0
i
=
m
,
n
I
(1)
i
̇
f
2
i
+
I
(2)
i
f
2
i
+
1
L
2
i
=
m
,
n
j
=
m
,
n
p
=
m
,
n
q
=
m
,
n
f
i
̇
f
j
f
p
̇
f
q
I
(3)
ijpq
f
i
f
j
f
p
f
q
I
(4)
ijpq
dt
,
(A16)
where
I
(1)
i
=

1
0
μ
2
i
d
ζ,
I
(2)
i
=

1
0
EI
L
4


i
2
d
ζ,
I
(3)
ijpq
=

1
0
μ


ζ
0


i


j
d
ξ

ζ
0


p


q
d
ξ


d
ζ,
I
(4)
ijpq
=

1
0
EI
L
4


i


j


p


q
d
ζ.
(A17)
The integrals in Eq. (
A17
) possess strong symmetries with respect to their indices. This property is used in the derivation
of Eq. (
A16
). The governing equations for
f
m
(
t
) and
f
n
(
t
) are calculated by finding the extremum of Eq. (
A16
) via the Euler-
Lagrange equations, yielding
I
(1)
m
f

m
(
t
)
+
I
(2)
m
f
m
(
t
)
+
1
L
2
g
mn
(
t
)
=
0
,
(A18)
with
g
mn
(
t
)
=
I
(3)
mmmm
f

m
(
t
)
+
I
(3)
mmmn
f

n
(
t
)
f
2
m
(
t
)
+

2
I
(3)
mmmn
f

m
(
t
)
+
I
(3)
mmnn
+
I
(3)
mnmn
f

n
(
t
)

f
m
(
t
)
f
n
(
t
)
+
I
(3)
mnmn
f

m
(
t
)
+
I
(3)
nnmn
f

n
(
t
)
f
2
n
(
t
)
+

I
(3)
mmmm

f

m
(
t
)

2
+
2
I
(3)
mmmn
f

m
(
t
)
f

n
(
t
)
+
I
(3)
mmnn

f

n
(
t
)

2

f
m
(
t
)
+

I
(3)
mmmn

f

m
(
t
)

2
+
2
I
(3)
mnmn
f

m
(
t
)
f

n
(
t
)
+
I
(3)
nnmn

f

n
(
t
)

2

f
n
(
t
)
+
2
I
(4)
mmmm
f
3
m
(
t
)
+
3
I
(4)
mmmn
+
I
(4)
mnmm
f
2
m
(
t
)
f
n
(
t
)
+
I
(4)
mmnn
+
4
I
(4)
mnmn
+
I
(4)
nnmm
f
m
(
t
)
f
2
n
(
t
)
+
I
(4)
mnnn
+
I
(4)
nnmn
f
3
n
(
t
)
,
(A19)
where the governing equation for
f
n
is obtained by swapping the
m
and
n
indices in Eqs. (
A18
) and (
A19
).
224303-10