Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances
Shock and Vibration
10709622
Nonlinear response of cantilever beams to combination and subcombination resonances
Ali H. Nayfeh 0
Haider N. Arafat 0
0 Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University , Blacksburg, VA 24061 , USA
Beams; combination resonance; parametric resonance; subcombination resonance; bifurcations

The nonlinear planar response of cantilever metallic beams
to combination parametric and external subcombination
resonances is investigated, taking into account the effects of
cubic geometric and inertia nonlinearities. The beams
considered here are assumed to have large lengthtowidth
aspect ratios and thin rectangular cross sections. Hence, the
effects of shear deformations and rotatory inertia are
neglected. For the case of combination parametric resonance, a
twomode Galerkin discretization along with Hamilton’s
extended principle is used to obtain two secondorder
nonlinear ordinarydifferential equations of motion and associated
boundary conditions. Then, the method of multiple scales is
applied to obtain a set of four firstorder nonlinear
ordinarydifferential equations governing the modulation of the
amplitudes and phases of the two excited modes. For the case
of subcombination resonance, the method of multiple scales
is applied directly to the Lagrangian and virtualwork term.
Then using Hamilton’s extended principle, we obtain a set of
four firstorder nonlinear ordinarydifferential equations
governing the amplitudes and phases of the two excited modes.
In both cases, the modulation equations are used to
generate frequency and forceresponse curves. We found that the
trivial solution exhibits a jump as it undergoes a subcritical
pitchfork bifurcation. Similarly, the nontrivial solutions also
exhibit jumps as they undergo saddlenode bifurcations.
1. Introduction
When a system is parametrically excited,
combination parametric resonances may occur when the
forcing frequency !i !j , where !k is the natural
frequency of the kth mode. When the excitation is direct,
an external combination resonance can occur in
systems with quadratic nonlinearities when !i !j
and in systems with cubic nonlinearities when
j!i !j !kj or j2!i !j j. An external
subcombination resonance can occur when a forcing
frequency is near onehalf the sum or difference of two or
more natural frequencies (Nayfeh and Mook [9]).
Dugundji and Mukhopadhyay [4] investigated the
response of a thin cantilever metallic beam to
combination parametric resonances involving the first
bending and torsional modes (i.e., !B1 + !T 1) in one
case and the second bending and first torsional modes
(i.e., !B2 + !T 1) in another. Their
experimental results show that the beam exhibits significant
oscillations both in bending and in torsion. In addition,
at large excitation amplitudes they observed the beam
snappingthrough and whipping around. Cartmell and
Roberts [3] theoretically and experimentally
investigated the stability of a cantilever beammass system
possessing the two simultaneous combination
parametric resonances !B1 + !T 1 !B2 !T 1.
They analyzed their system using the method of
multiple scales and found good agreement between
theory and experiment within certain ranges of the
excitation frequency. However, in other regions where
periodic modulations can occur, the correlation was not
satisfactory because the theoretical solution could not
predict nonstationary responses.
Kar and Sujata [5] investigated the instability of an
elastically restrained cantilever beam subjected to
uniaxial and follower forces. They found that combination
parametric resonances of the difference type do not
occur when the force is uniaxial or supertangential, but
that they are predominant when the force is tangential
or subtangential. Kar and Sujata [6] also investigated
the instability of a rotating, pretwisted, and preconed
cantilever beam, taking into consideration the
Coriolis effects. They found that the Coriolis force may
increase the instability regions in the case of combination
parametric resonances.
Anderson et al. [
1
] experimentally investigated the
response of a thin metallic cantilever beam with an
initial curvature to a combination parametric excitation.
The first four natural frequencies are 0.65 Hz, 5.65 Hz,
16.19 Hz, and 31.91 Hz. They found that, over a range
of forcing frequency above 32 Hz, the first and fourth
modes are activated by a combination parametric
resonance with the first mode dominating the response.
Sridhar et al. [12] investigated the response of a
hingedclamped beam to the subcombination
resonance 12 (!a !b) and the combination
resonance !a !b !c. Yamamoto et al. [16,17]
theoretically and experimentally investigated the
nonlinear response of simplysupported beams to
combination and subcombination resonances, respectively.
They [16] found that, in order to excite the external
combination resonance, one needs a timeindependent
component in the excitation. However, they [17] found
that the external subcombination resonance can be
excited with only a harmonic excitation. In both cases,
they found that only additivetype resonances can be
activated. In these three studies, nonlinearities due to
midplane stretching were included in the analysis.
The experimental results of Dugundji and
Mukhopadhyay [4] and Anderson et al. [
1
] confirm the
occurrence of such resonances in structures. More
important, their results demonstrate that such resonances
can be a mechanism where a highfrequency excitation
can activate lowfrequency largeamplitude modes. For
example, the ratio of the excitation frequency to the
natural frequency of lowest mode excited was
approximately 18 : 1 in the experiments of Dugundji and
Mukhopadhyay [4] and 49 : 1 in the experiments of
Anderson et al. [
1
]. The analyses of Cartmell and
Roberts [3] and Kar and Sujata [5,6] did not take into
consideration the effect of nonlinearities inherent in the
system.
In this paper, we investigate the response of a
uniform thin metallic cantilever beam to either a
combination parametric resonance or a subcombination
resonance of two modes (see Fig. 1). Because such
resonance phenomena cannot be adequately explained by
using linear theories of vibrations, it is necessary to
incorporate the effects of nonlinearities in the analysis.
Furthermore, because the presence of a lowfrequency
component in the response may cause the beam to
oscillate with large amplitudes, we account for both
geometric and inertia nonlinearities. The method of
multiple scales is used to determine two sets of four
firstorder nonlinear ordinarydifferential equations
governing the modulation of the amplitudes and phases of the
two interacting modes. The modulation equations are
then used to generate frequency and forceresponse
curves.
2. Combination parametric resonance
The nondimensional equation of motion for
inextensional cantilever beams where the effects of shear
deformation and rotatory inertia are neglected is given by
where the dimensional time t = tpmL4=EI and the
dimensional deflection and arclength are v = Lv and
s = Ls. The boundary conditions are
v + cv_ + viv =
0
+ F (s, v) cos( t),
v02 ds ds
0
v = 0 and v0 = 0 at s = 0,
v00 = 0 and v000 = 0 at s = 1:
The corresponding nondimensional Lagrangian and
virtual work are given by
0
L =
1 1 @ Z s
1 v_ 2 +
2 2 2 @t 0
v02 ds
2
1 v002 + v02v002
2
ds,
W =
Qv v ds
(1)
(2)
(3)
(4)
+
+
where the prime denotes differentiation with respect to
the arclength s and the dot denotes differentiation with
respect to time t. Eqs (1)–(5) are valid for beams that
are uniform, homogeneous, long, and thin. For stubby
or thick beams, shear deformation and rotatory inertia
effects may not be negligible (Timoshenko [14]).
In the presence of damping, all of the modes that are
not directly excited or indirectly excited by an internal
resonance will decay with time. Hence, for the case of
combination parametric resonance or external
subcombination resonance of the mth and nth modes, where
the i are the orthonormal mode shapes, the longtime
response of the beam will consist only of these two
modes if neither of them is involved in an internal
resonance with any other mode. Therefore, we assume a
solution for v in the form
v(s, t) =
m(s) m(t) +
n(s) n(t):
For cantilever beams,
i(s) = ci cosh(zis)
cos(zis)
+
cos(zi) + cosh(zi)
sin(zi) + sinh(zi)
sin(zis)
sinh(zis) ,
where zi is the ith root of 1 + cos(z) cosh(z) = 0 and
ci is chosen so that R01 i2 ds = 1. The nondimensional
natural frequencies are given by
2
!i = zi :
The first four nondimensional frequencies are !1 =
3:5160, !2 = 22:0345, !3 = 61:6972, and !4 =
120:9019.
For the case of combination parametric resonance,
we let
F (s, v) =
v00(s
1) + v0 f:
Substituting Eqs (6)–(9) into Eqs (4) and (5) and
integrating the result over space, we obtain the discretized
Lagrangian and virtual work as
1
L = 2
1 + 1 m2 + 2 2 m n + 3 n2 _m2
(6)
(7)
(8)
(9)
= Qm,
we obtain
2
m + 2 m _m + !m m
=
4 1 m3 + 3 2 m n + 2 3 m n2 + 4 n
2 3
1 m2 + 2 2 m n + 3 n2 m
7 m2 + 8 m n + 9 n2 n
1 m + 2 n _m2
8
4
m + 2 9
2
5 n _n
2 2 m + 3 n _m _n
fmm m + fmn n cos( t),
(14)
n + 2 n _n + !n2 n
=
2 m3 + 2 3 m n + 3 4 m n2 + 4 5 n
2 3
7 m2 + 8 m n + 9 n2 m
4 m2 + 2 5 m n + 6 n2 n
To determine a secondorder uniform expansion for
the solutions of Eqs (14) and (15) for the case of
com= !m + !n + "2 ,
where " is a small nondimensional bookkeeping
parameter. Next, using the method of multiple scales
(Nayfeh [7]), we obtain
m = " Am(T2)ei!mT0 + Am(T2)e i!mT0 +
n = " An(T2)ei!nT0 + An(T2)e i!nT0 +
,(17)
, (18)
where T0 = t, T2 = "2t, and Am and An are governed
by
The Sij and fij were calculated for combination
parametric resonances of the additive type for
different pairs of the first four modes. The results are shown
in Table 1. It follows from Table 1 that S11 > 0 and
S22, S33, and S44 < 0. Hence, the nonlinearity is of the
hardening type for the first mode and of the softening
type for the higher modes.
The complexvalued modulation equations (19) and
(20) can be transformed into a realvalued form by
introducing the transformation
1
Am =
2
amei m
and
1
An = 2 anei n :
(22)
Substituting Eqs (22) into Eqs (19) and (20) and
separating real and imaginary parts, we obtain
a0m =
mam
fmn an sin ,
4!m
am m0 = Smm a3m + Smn
8!m 8!m
ama2n
+ fmn an cos ,
4!m
a0n =
nan
fnm am sin ,
4!n
an n0 = S8!nmn a2man + 8S!nnn a3n + f4n!mn am cos , (26)
where
(16)
(19)
(20)
(23)
(24)
(25)
(27)
(28)
bination parametric resonance of the additive type, we
scale i and fij as "2 i and "2fij and introduce the
detuning parameter so that
T2
m
n:
Substituting Eqs (22) into Eqs (17) and (18) and then
substituting the result into Eq. (6), we find that the
beam response is given by
v(s, t)
" am m(s) cos !mt +
m
+ an n(s) cos !nt + n ,
where the ai and i are given by Eqs (23)–(27).
Using Eqs (16) and (27) to eliminate !n and n from
Eq. (28), we have
v(s, t)
" am m(s) cos !mt +
+ an n(s) cos
!m t
m
m
: (29)
The equilibrium solutions or fixed points of
Eqs (23)–(27) correspond to a0m = 0, a0n = 0, and 0 =
0, which in turn correspond to twoperiod
quasiperiodic responses of the beam according to Eq. (29).
There are two possible equilibrium solutions: (a) am =
0 and an = 0 and the beam is not excited and (b) am 6=
0 and an 6= 0 and the beam response is quasiperiodic.
In the latter case, Eqs (24), (26), and (27) can be used
to eliminate m and n to obtain the following
equation for :
0 =
Smm + Snm
8!m 8!n
a2m
anfmn + amfnm
4am!m 4an!n
Smn + Snn
8!m 8!n
cos :
a2n
(30)
Thus, for nontrivial solutions, the modulation
equations are reduced from four to three firstorder
differential equations. For equilibrium solutions, we set the
time derivatives in Eqs (23), (25), and (30) equal to
zero and solve for am, an, and , yielding the
following closedform solution:
m +
p
m n
n r fmnfnm
16!m!n
m n,
ea2m =
a2n =
sin
=
where
= 4
The stability of a nontrivial equilibrium solution can
then be studied by calculating the eigenvalues of the
Jacobian matrix of Eqs (23), (25), and (30) evaluated
at this equilibrium solution.
To determine the stability of the trivial
equilibrium solutions, we study the stability of the linearized
complexvalued modulation equations (19) and (20).
To this end, we let
Am = cme T2+i T2
and
in the linearized equations (19) and (20) and obtain
It follows from Eqs (35) that the trivial solution is
stable if the real parts of both ’s are negative.
In Fig. 2, we show typical frequencyresponse curves
for a combination parametric resonance of the additive
type of the first two modes when the excitation
amplitude is f = 10. Clearly, the first mode dominates
the response. Although the nonlinearity is hardening
for the first mode and softening for the second mode,
the frequencyresponse curves are bent to the left,
indicating a softening behavior for both modes. This is
so because
according to Eq. (24). Although S11 is positive, S12
is negative and its magnitude is much larger than S11.
Hence, the nonlinearity decreases the frequency of the
first mode, and hence bends the frequencyresponse
curves to the left. It follows from Fig. 2 that,
depending on how is varied, the trivial solution loses
stability via either a subcritical or a supercritical pitchfork
bifurcation.
In Fig. 3, we show amplituderesponse curves for a
combination parametric resonance of the additive type
of the first two modes. In part (a), the frequency
detuning parameter = 1, and in part (b) = 1. When
= 1, there are two branches of nontrivial
fixedpoint solutions, one stable and the other unstable. As f
is increased away from zero, the trivial solution loses
stability via a subcritical pitchfork bifurcation, causing
the response to jump up to the stable branch of
nontrivial solutions. Similarly, a fixedpoint on the stable
nontrivial branch loses stability via a saddlenode
bifurcation as f is decreased, resulting in a jump down to the
trivial branch. When > 0, there are only branches
of stable nontrivial fixed points, as shown in Fig. 3(b).
The nontrivial solution is activated gradually as the
trivial solution undergoes a supercritical pitchfork
bifurcation.
In Figs 4 and 5, the frequency and
amplituderesponse curves are presented when the first and fourth
modes are activated by the combination parametric
resonance. The forcing amplitude in Fig. 4 is f = 10 and
the detuning parameter is = 1 in Fig. 5(a) and
= 1 in Fig. 5(b). We note that the behaviors in Figs 4
and 5 are similar to those in Figs 2 and 3. However, the
amplitudes when modes 1 and 4 are excited are about
an order of magnitude smaller than those when modes
1 and 2 are excited.
Anderson et al. [
1
] experimentally investigated the
response of a cantilever beam where 2!3
!1 + !4. They found that over a small region of
frequency detuning, only the first and fourth modes were
excited by a combination parametric resonance. The
results shown in Fig. 4 agree qualitatively with their
frequencyresponse curves.
Results for the case of a combination parametric
resonance of the difference type can be obtained by
replacing !m by !m and m by m in Eqs (23)–(32).
However, it can be seen from Eq. (32) that this
resonance cannot be activated in this system.
3. External subcombination resonance
In this section, we consider the response of the beam
to the subcombination resonance 12 (!n !m).
In this case, the excitation, which is transverse, is
assumed to be hard. Therefore, we let F (s, v) = "f (s)
in Eqs (1) and (5). Furthermore, in order that the
cubic nonlinearities and damping balance the resonance,
we scale c as "2c. We use the method of timeaveraged
Lagrangian and virtual work to determine a uniform
firstorder expansion. To this end, we let
v(s, T0, T2)
" Am(T2) m(s)ei!mT0
+ An(T2) n(s)ei!nT0 +
(s)ei T0 + cc , (40)
where
c1 =
c3
f
= 4 2
f
c2 = 4 2
sin p
1 + cos p
where m(s) and n(s) are the mode shapes
corresponding to the natural frequencies !m and !n and
(s) is governed by the boundaryvalue problem
iv
2
and
We note that 2" (s) cos( t) is the particular solution
of the linear undamped beam equation and associated
boundary conditions. When f (s) is constant, the
solution of Eqs (41) and (42) can be expressed as
where the Sij are defined in Eqs (21), the i are
defined in Appendix A, and m, n, and are defined in
Appendix B. In Table 2, we present the numerical
values for the coefficients m, n, and for external
subcombination resonances of the additive type for
different pairs of the first four modes. Applying Hamilton’s
principle to Eqs (48) and (49), we obtain the
modulation equations
Substituting the polar transformation, Eqs (22), into
Eqs (50) and (51) and separating real and imaginary
AmA0m
AnA0n
SmmA2mAm
2
SmnAmAmAnAn
mAmAm
parts, we obtain the realvalued modulation equations
a0m =
mam
an sin ,
m am +
2!m
+ Smn
8!m
2!m
Smm 3
am
8!m
ama2n +
an cos ,
2!m
am sin ,
a0n =
nan
an n0 =
2!n
n an + Snm a2man
2!n 8!n
+ Snn a3n +
8!n
2!n
am cos ,
Values of the coefficients
tions of the first four modes
where
(57)
which is periodic having the same period as that of the
excitation. In this case, the external subcombination
resonance is not activated. The stability of this
trivial solution can be analyzed by investigating solutions
of the linearized complexvalued modulation equations
(50) and (51). To this end, we let
Am = cme T2+2i T2
and
in the linearized equations (50) and (51) and obtain
h
2i!m
2 +
+
m +
cm +
+ 2i
h
2i!n
+
i
m cm +
n +
+
cn = 0,
i
n cn = 0:
(59)
(60)
For nontrivial solutions,
m +
n
+ i 2
m n
!n
n +
m n
4!m!n
m n +
2!m
n m
2!n
m +
2!m
It follows from Eqs (58) that the trivial solution loses
stability as one of the ’s crosses the imaginary axis
along the real axis from the lefthalf to the righthalf of
the complex plane.
For nontrivial solutions, we use Eqs (53), (55), and
(56) to eliminate m and n and obtain
0 = 2
where
In Fig. 6, we show typical amplituderesponse curves
for the subcombination external resonance of the first
two modes for = 1. The trivial solution loses
stability via a subcritical pitchfork bifurcation as the
forcing amplitude is increased, resulting in a jump in the
response amplitudes. On the other hand, as the
forcing amplitude is decreased from a large value, the
trivial solution loses stability through a supercritical
pitchfork bifurcation, resulting in a gradual increase in the
response amplitudes. In either case, the nontrivial
solution loses stability as f is decreased via a
saddlenode bifurcation. Comparing Figs 3 and 6, we
conclude that the linear shift in the natural frequencies
i=2!i and the nonlinear dependence of the
effective forcing ( / f 2) on the excitation amplitude
have dramatic qualitative and quantitative effects on
the forceresponse curves.
In Fig. 7, we show typical frequencyresponse curves
for the same resonance when f = 20. As in the case of
combination parametric resonance, the curves are bent
to the left, indicating a softeningtype nonlinearity.
Because 1 and 2 are negative and proportional to f 2,
there is a strong decrease in the linear natural
frequencies with an increase in f . Consequently, for f = 20,
unlike the combination parametric resonance, the
external subcombination resonance is activated only for
negative values of . We also note that increasing the
forcing amplitude causes both the stable and unstable
branches to shift to the left, with the latter being shifted
more than the former.
In Fig. 8, we show typical amplituderesponse curves
for a subcombination external resonance of the first
and third modes for = 1. Comparing Figs 6 and
8, we note that the amplituderesponse curves for the
external subcombination resonance of modes 1 and 3
are qualitatively different from the amplituderesponse
curves for the external subcombination resonance of
modes 1 and 2. As in Fig. 6, the trivial solution in Fig. 8
loses stability via a subcritical pitchfork bifurcation as
f is increased, resulting in a jump in the response
amplitudes. However, the amplitudes of the nontrivial
solutions in Fig. 8 increase as f is increased, in contrast
to the results in Fig. 6, where the amplitudes of the
nontrivial solutions decrease as f is increased.
Comparing Figs 3(a) and 5(a) with Fig. 8, we note
that the amplituderesponse curves for the external
subcombination resonance of modes 1 and 3 are
similar to those obtained for the combination parametric
resonance. Therefore, the effects of the linear shifts in
the natural frequencies and the nonlinear dependence
of the effective forcing on the excitation amplitude
f do not change qualitatively the amplituderesponse
curves.
In Fig. 9, we show typical frequencyresponse curves
for the external subcombination resonance of modes 1
and 3 when f = 50. Again the curves are bent to the
left, indicating that the nonlinearity and the linear shift
1=2!1 decrease the frequency of the dominant first
mode. Furthermore, similar to the combination
parametric resonance, this case of external subcombination
resonance may be activated for positive as well as
negative values of .
Of the cases mentioned in Table 2, we found that
the responses obtained for the external subcombination
resonance of modes 2 and 4 are qualitatively similar
to those obtained for the external subcombination
resonance of modes 1 and 3, whereas the behaviors of the
remaining cases are qualitatively similar to the external
subcombination resonance of modes 1 and 2.
Finally, we note again that the case of external
subcombination resonance of the difference type can be
studied by replacing !m by !m and m by m in
Eqs (52)–(56). However, it can be seen from Eq. (64)
that this resonance cannot be activated.
4. Conclusion
The nonlinear flexural responses of cantilever beams
to combination parametric and subcombination
resonances have been investigated. For the case of
combination parametric resonance, the beam is excited
longitudinally, whereas for the case of external
subcombination resonance, the beam is excited transversely. In
the parametric case, the Lagrangian and virtualwork
term are discretized using a twomode Galerkin
technique and Hamilton’s extended principle is used to
obtain two secondorder nonlinear ordinarydifferential
equations of motion. Then, the method of multiple
scales is used to obtain a set of four firstorder
nonlinear ordinarydifferential equations governing the
modulation of the amplitudes and phases of the two
excited modes. In the subcombination case, the method
of timeaveraged Lagrangian and virtual work along
with Hamilton’s extended principle are used to obtain
the modulation equations.
We found that the excitation amplitude must exceed
a certain threshold for either resonance to be activated.
For the external subcombination resonance, two
qualitatively different amplituderesponse behaviors were
found. In the first, the external subcombination
resonance will not be activated if the excitation amplitude
is chosen beyond a certain limit. In the second,
similar to the case of combination parametric resonance, no
upper limit on the excitation amplitude exists for the
resonance to be activated.
In both parametric combination and external
subcombination resonances, the trivial solution loses
stability via pitchfork bifurcations, both supercritical and
subcritical, thereby producing nontrivial responses.
When the pitchfork bifurcation is supercritical, the
change in amplitudes is gradual and therefore the
transition is smooth. When the pitchfork bifurcation is
subcritical, the change in amplitudes is abrupt and is
associated with a jump. In addition, the nontrivial solutions
lose stability via saddlenode bifurcations as the
excitation amplitude is decreased below a critical value,
resulting in a jump down to the trivial solution.
For cantilever beams, we found that combination
parametric and external subcombination resonances of
the difference type cannot be activated. Rather, only
additivetype resonances can be excited.
Acknowledgment
This work was supported by the National Science
Foundation under Grant No. CMS9423774.
Appendix A
1 =
Z s
0
Z 1
0
0
Z 1
0
Z s
0m2 ds
Z s
0
Z s
0
Z s
0
0n2 ds
Z s
0
2
ds,
0m2 ds
ds,
0m2 ds
Z s
ds,
4 =
5 =
i =
2
Appendix B
i =
Z s
0
2
0 0 ds
i
ds,
!n
0n 0 ds
02 ds
ds:
[2] K.G. Asmis and W.K. Tso, Combination and internal resonance
in a nonlinear twodegreesoffreedom system, J. Applied
Mechanics, Trans. ASME 39 (1972), 832–834.
[3] M.P. Cartmell and J.W. Roberts, Simultaneous combination
resonances in a parametrically excited cantilever beam, Strain
23 (1987), 117–126.
+ 2 0i2 002
2 !i2 +
2
+ 0m 0n 002 +
Z s
0
Z s
0
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[1] T.J. Anderson , B. Balachandran and A.H. Nayfeh , Nonlinear resonances in a flexible cantilever beam , J. Vibration and Acoustics, Trans. ASME 116 ( 1994 ), 480  484 .