Dynamics of a Cubic Nonlinear Vibration Absorber

TLDR: In this article, the authors considered a plant model possessing curvature and inertia nonlinearities and introduced a second-order absorber that is coupled with the plant through user-defined cubic nonlinearity.

Abstract: We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.

Figures

Table 2. Values of the ratior on branchAB.

Figure 6. Time evolution of the plant amplitude whenδ = 1, µ1 = 0.0005,µ2 = 0.001, ν1 = 0.01, and f = 0.0001. (a)σ2 = 0.003878056, (b)σ2 = 0.00387801, (c)σ2 = 0.0038775, and (d)σ2 = 0.003876.

Figure 1. Frequency-response curves whenµ1 = 0.0005,µ2 = 0.001,ν1 = 0.01, δ = −1, andf = 0.0001. (a) Overall frequency-response curves and (b) an enlargement of area ‘I’. Solid (dotted) lines denote stable nodes (saddles).

Figure 5. Time responses whenδ = 1, µ1 = 0.0005, µ2 = 0.001, ν1 = 0.01, f = 0.0001, and σ2 = 0.001740453. Initial conditions on the unstable manifold of branchAB.

Table 1. Possible feedback signals.

Figure 8. Projection of the Hopf bifurcation loci on theδ–f plane whenµ1 = 0.0005 andµ2 = 0.001. (a)σ2 = −0.005 and (b)σ2 = 0.005. Solid (dotted) lines denote supercritical (subcritical) bifurcations.

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Distributed under a Creative Commons Attribution| 4.0 International License
Dynamics of a Cubic Nonlinear Vibration Absorber
Shac S. Oueini, Char-Ming Chin, Ali H. Nayfeh
To cite this version:
Shac S. Oueini, Char-Ming Chin, Ali H. Nayfeh. Dynamics of a Cubic Nonlinear Vibration Absorber.
Nonlinear Dynamics, Springer Verlag, 1999, 20 (3), pp.283-295. �10.1023/A:1008358825502�. �hal-
01580924�

Dynamics of a Cubic Nonlinear Vibration Absorber
SHAFIC S. OUEINI, CHAR-MING CHIN, and ALI H. NAYFEH
Department of Engineering Science and Mechanics (MC 0219), Virginia Polytechnic Institute and State
University, Blacksburg, VA 24061, U.S.A.
Abstract. We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing
curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through
user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is
approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the
forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode
(plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We
obtain an approximate solution to the governing equations using the method of multiple scales and show that the
system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and
demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic
responses.
Keywords: Vibration absorber, saturation, internal resonance, bifurcations.
1. Introduction
Nonlinearities are responsible for unusual phenomena in the presence of internal and/or ex-
ternal resonances. Of particular interest are systems coupled with quadratic nonlinearities and
possessing a two-to-one internal resonance. Sethna [1] was one of the first researchers to study
such systems. He conducted theoretical studies, performed analog simulations, and showed
that nonperiodic motions may exist. Theoretical and experimental studies by Nayfeh et al. [2],
Haddow et al. [3], and Balachandran and Nayfeh [4] on L-shaped structures have shown that,
when two degree-of-freedom systems coupled with quadratic nonlinearities possess a two-to-
one internal resonance and the higher mode is subjected to a primary resonance, there exists
a saturation phenomenon. When the forcing amplitude exceeds a certain threshold, the amp-
litude of the directly excited mode remains constant, and the excitation energy is channeled to
the unexcited lower mode. As the forcing amplitude increases, the response amplitude of the
lower mode increases, while the response amplitude of the higher mode saturates. Bajaj et al.
[5] and Banerjee et al. [6] investigated the response of a pendulum mounted on an oscillating
mass. They used first- and second-order averaging methods to analyze the dynamics of the
system. In addition to reporting the occurrence of periodic and chaotic motions, they found
that the saturation phenomenon predicted by the first-order averaging technique is lost when
the effect of higher-order nonlinearities is included in the model.
Recently, Oueini et al. [7] and Pratt et al. [8] exploited the saturation phenomenon in
devising an active vibration suppression technique. They introduced a second-order absorber
and coupled it with the plant through a user-defined quadratic feedback control law. Once the
1

Table 1. Possible feedback signals.
Sensor F
f
(t) F
c
(t)
Position u
1
u
2
2
, ¨u
1
u
2
2
u
2
u
2
1
, u
2
˙u
2
1
, u
2
¨u
2
1
, u
2
u
1
¨u
1
Velocity u
1
˙u
2
2
, ¨u
1
˙u
2
2
˙u
2
u
1
˙u
1
, ˙u
2
˙u
1
¨u
1
Acceleration u
1
¨u
2
2
, ¨u
1
¨u
2
2
¨u
2
u
2
1
, ¨u
2
˙u
2
1
, ¨u
2
¨u
2
1
, ¨u
2
u
1
¨u
1
coupling between the plant and the absorber is established through a sensor and an actuator,
effective vibration suppression is achieved by tuning the natural frequency of the absorber to
one-half the excitation frequency.
Unlike previous studies that investigated the saturation phenomenon in quadratic systems,
we propose to ‘reverse-engineer’ the saturation phenomenon using cubic terms. We consider
a plant modeled by a second-order nonlinear differential equation and introduce an active
vibration absorber coupled with the plant via a specific set of cubic nonlinearities. We analyze
the resulting equations using the method of multiple scales and show that a saturation phe-
nomenon occurs when a one-to-one internal resonance is imposed between the plant and the
absorber. To our knowledge, this is the first instance in which the saturation phenomenon is
encountered in cubicly coupled systems.
2. System Model and Perturbation Solution
The plant is a cantilever beam whose response is governed by a nonlinear partial-differential
equation. We consider a mode that is not involved in an internal resonance with any of the
other modes. Then, application of a single-mode discretization scheme yields the ordinary-
differential equation
¨u
2
+ 2ε
2
˜µ
2
˙u
2
+ ω
2
2
u
2
+
˜
δ
1
u
3
2
+
˜
δ
2
¨u
2
u
2
2
+
˜
δ
3
˙u
2
2
u
2
= ε
3
F cos(t) +˜α
2
F
c
(t), (1)
where u
2
is the generalized coordinate of the mode under consideration, ˜µ
2
is a damping
coefficients, ω
2
is the natural frequency, the
˜
δ
i
are constants, F and  are the forcing amp-
litude and frequency, respectively, ˜α
2
is a constant gain, F
c
(t) is a control signal, and ε is a
dimensionless bookkeeping parameter. The model includes the curvature nonlinearity u
3
and
the inertia nonlinearities ¨u
2
u
2
2
and ˙u
2
2
u
2
.
We introduce a second-order absorber and couple it with the plant through a user-defined
cubic feedback control law. Then, the equation governing the dynamics of the absorber is
given by
¨u
1
+ 2ε
2
˜µ
1
˙u
1
+ ω
2
1
u
1
=˜α
1
F
f
(t), (2)
where u
1
is the absorber coordinate and ˜µ
1
and ω
1
are the absorber’s damping coefficient and
frequency, respectively, ˜α
1
is a constant gain, and F
f
(t) is a feedback signal. The feedback
and control signals may take different forms depending on the available sensor. We list all
possible combinations in Table 1. Furthermore, we choose the absorber’s frequency such that
ω
1
≈ ω
2
(i.e., one-to-one internal resonance).
2

We consider the case of primary resonance (i.e.,  ≈ ω
2
) and position feedback and
analyze, without loss of generality, the system of equations
¨u
1
+ 2ε
2
˜µ
1
˙u
1
+ ω
2
1
u
1
= α
1
¨u
1
u
2
2
, (3)
¨u
2
+ 2ε
2
˜µ
2
˙u
2
+ ω
2
2
u
2
+
˜
δ
1
u
3
2
+
˜
δ
2
¨u
2
u
2
2
+
˜
δ
3
˙u
2
2
u
2
= α
2
˙u
2
1
u
2
+ ε
3
F cos(t). (4)
Using the method of multiple scales [9], we obtain an approximate solution to Equations (3)
and (4) in the form
u
1
≈ A
1
(T
2
) e
iω
1
T
0
+ A
1
(T
2
) e
−iω
1
T
0
, (5)
u
2
≈ A
2
(T
2
) e
iω
2
T
0
+ A
2
(T
2
) e
−iω
2
T
0
, (6)
where T
0
= t, T
2
= ε
2
t,and
2i

dA
1
dT
2
+˜µ
1
A
1

+ 8ˆα
1
(2A
2
A
2
A
1
+ A
1
A
2
2
e
2i ˜σ
1
T
2
) = 0, (7)
2i

dA
2
dT
2
+˜µ
2
A
2

+ 8δ
e
A
2
2
A
2
+ 8ˆα
2
(−2A
1
A
1
A
2
+ A
2
1
A
2
e
−2i ˜σ
1
T
2
) − f e
i ˜σ
2
T
2
= 0. (8)
Here,
ˆα
1
=
1
8
ω
1
α
1
, ˆα
2
=
1
8
ω
1
α
2
, ˜σ
1
T
2
= (ω
2
− ω
1
)T
0
, ˜σ
2
T
2
= (ω
2
− )T
0
,
δ
e
=
1
8ω
2
h
3
˜
δ
1
− ω
2
2
(3
˜
δ
2
+
˜
δ
3
)
i
and f =
F
2ω
2
.
To facilitate the analysis, we reduce the number of the parameters in Equations (7) and (8)
by introducing the scalings
A
1
= c
1
B
1
,A
2
= c
2
B
2
, and T
2
= c
3
T, (9)
where the c
i
are constants. Then, Equations (7) and (8) become
2i(B
0
1
+ µ
1
B
1
) + 8ˆα
1
c
2
2
c
3
(2B
2
B
2
B
1
+ B
1
B
2
2
e
2iσ
1
T
) = 0, (10)
2i(B
0
2
+ µ
2
B
2
) + 8c
2
2
c
3
δ
e
B
2
2
B
2
+ 8ˆα
2
c
2
1
c
3
(−2B
1
B
1
B
2
+ B
2
1
B
2
e
−2iσ
1
T
) −
c
3
f
c
2
e
iσ
2
T
= 0, (11)
where
µ
i
= c
3
˜µ
i
and σ
i
= c
3
˜σ
i
, (12)
and the prime represents differentiation with respect to the time variable T . To keep the forcing
amplitude f as a bifurcation parameter, we set c
2
= c
3
. Furthermore, we let
ˆα
1
c
3
3
= 1andˆα
2
c
2
1
c
3
= 1. (13)
3

Solving for the constants, we obtain
c
1
=
s
|ˆα
1
|
1/3
|ˆα
2
|
and c
2
= c
3
=|ˆα
1
|
−1/3
. (14)
Moreover, we define
δ =
δ
e
ˆα
1
. (15)
Next, we express B
1
(T ) and B
2
(T ) in the polar form
B
1
=
1
2
a
1
(T ) e
iβ
1
(T )
and B
2
=
1
2
a
2
(T ) e
iβ
2
(T )
. (16)
Substituting Equations (14–16) into Equations (10) and (11) and separating real and imaginary
parts yields
a
0
1
=−µ
1
a
1
− a
1
a
2
2
sin θ
1
, (17)
a
0
2
=−µ
2
a
2
+ a
2
1
a
2
sin θ
1
+ f sin θ
2
, (18)
a
1
β
0
1
= a
1
a
2
2
(2 + cos θ
1
), (19)
a
2
β
0
2
= δa
3
2
+ a
2
1
a
2
(cos θ
1
− 2) − f cos θ
2
, (20)
where
θ
1
= 2(β
2
− β
1
+ σ
1
T) and θ
2
= σ
2
T − β
2
. (21)
3. Equilibrium and Dynamic Solutions
In this section, we study the equilibrium and dynamic solutions of Equations (17–21) and their
bifurcations. To determine the equilibrium solutions, we set a
0
1
= a
0
2
= 0andθ
0
1
= θ
0
2
= 0and
obtain the algebraic equations
µ
1
a
1
=−a
1
a
2
2
sin θ
1
, (22)
µ
2
a
2
= a
2
1
a
2
sin θ
1
+ f sin θ
2
, (23)
ν
1
a
1
= a
1
a
2
2
(2 + cos θ
1
), (24)
ν
2
a
2
= δa
3
2
+ a
2
1
a
2
(cos θ
1
− 2) − f cos θ
2
, (25)
where
ν
1
= σ
1
+ σ
2
and ν
2
= σ
2
. (26)
There are two possibilities: a
1
= 0anda
1
6= 0. When a
1
= 0, it follows from Equations (23)
and (25) that
[µ
2
2
+ (ν
2
− δa
2
2
)
2
]a
2
2
− f
2
= 0. (27)
4

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Dynamics of a cubic nonlinear vibration absorber" ?

The authors study the dynamics of a nonlinear active vibration absorber. The authors consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, the authors show the existence of a saturation phenomenon. The authors obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, the authors perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.