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Accepted Manuscript
Cubic Quintic Septic Duffing Oscillator: An analytical study
S.K. Remmi, M.M. Latha
PII:
DOI:
Reference:
S0577-9073(18)30818-9
https://doi.org/10.1016/j.cjph.2018.08.009
CJPH 606
To appear in:
Chinese Journal of Physics
Received date:
Revised date:
Accepted date:
13 June 2018
6 August 2018
12 August 2018
Please cite this article as: S.K. Remmi, M.M. Latha, Cubic Quintic Septic Duffing Oscillator: An analytical
study, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.08.009
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ACCEPTED MANUSCRIPT
Highlights
• A three-spring structure is considered and its behaviour is analysed.
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• The dynamics is found to be governed by a Cubic Quintic Septic Duffing
equation.
• Linearised Harmonic Balance Method is used to solve the resulting
equation.
• Low Natural frequency is observed around the equilibrium point due
to low stiffness.
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• The result is compared with Cubic Quintic Duffing equation.
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S. K. Remmia , M. M. Lathab
a
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Cubic Quintic Septic Duffing Oscillator: An analytical
study
AN
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Register No: 12585, Department of Physics, Women’s Christian College, Nagercoil 629 001, India (Affiliated to Manonmaniam Sundaranar University, Abishekapatti,
Tirunelveli- 627 012, Tamilnadu, India)
b
Associate Professor, Department of Physics, Women’s Christian College, Nagercoil 629 001, India
Abstract
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This paper deals with a particular arrangement of a statically balanced system using 3 springs of prescribed material stiffness and critical geometrical
parameter. The dynamics is described by a nonlinear differential equation
upto septic power following odd nonlinearity for small disturbance from static
equilibrium position. The governing differential equation is solved analytically by the combination of the linearization of the equation with the method
of Harmonic Balancing to observe the low natural frequency at fixed point
and a finite displacement range in the neighbourhood of the equilibrium point
where the dynamic stiffness is low. By this approximation method, the behaviour of the displacement with increase in time as well as the phase-plot of
Cubic Quintic Septic Duffing equation for a set of parameter values is studied
at the equilibrium position and its neighbourhood.
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1. Introduction
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The way to interpret an object before performing real experiments is
through modelling. It is a mathematical representation which allows simulation and analysing about the system. For physical understanding a simple
analytic model is sufficient to describe significant properties of a natural
object. In this deem, mechanical systems that has been designing using
∗
Tel:+91 4652 - 231461; Fax:+91 4652 - 228834
Email address: lathaisaac@yahoo.com (M. M. Latha)
Preprint submitted to Chinese Journal of Physics
August 14, 2018
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spring-mass would be the most common in engineering applications. We are
recurrently familiar with single and two spring kind of models that have good
understanding in their behaviours. But analysing the dynamics of mass with
three springs typically arranged nonlinearly is a fascinating model in practice.
Recently, this configuration is widely used in vibration isolators under specified condition. The three-spring elastic model has the property to deform
with zero stiffness where any change in configuration requires no external
work [1]. The configuration is attained by appropriate combination of geometry, stiffness and prestress. The elastic structure typically designed with
suitable system parameters can establish a concept of neutrally stable configuration called Quasi zero stiffness (QZS) which is ultimately either stable
or unstable for small variations around the equilibrium state. This fascinating phenomenon maintains high static stiffness and low dynamic stiffness
at equilibrium and thus yields low natural frequency especially in passive
vibration isolators. This QZS behaviour near the operating region can be
effectively obtained by means of both geometrical and physical nonlinearity
of the springs. This consequently admits the system to exhibit nonlinear
behaviour. Carrella [2] studied the geometrical nonlinear three spring structure designed using negative stiffness element which turned out to be the
prominent feature of QZS mechanism and in addition there is an optimum
geometry and stiffness relationship to achieve stable QZS property in the
structure. As well a small change in stiffness around the equilibrium point
is also studied by introducing physical nonlinearity to the system by means
of the initial geometry, pre-stress and the stiffness of the springs [3]. The
dynamics of QZS system under small or large displacement from the static
equilibrium position may be linear or nonlinear and its strength of nonlinearity depends on the nonlinear spring force acting on the mass. The quest
for exact solution of such nonlinear systems supports to evolve number of
analytical techniques. In general, it is impossible to obtain an accurate solution for a strong nonlinear differential equation [4]. However, a well-known
Perturbation method [5,6] has been used in the past few decades for solving the nonlinear systems which hold only small parameters in its equation
of motion. But at present state of knowledge an approximate analytical
approach is applicable to solve the nonlinear differential equation contains
large as well as small parameters. Subsequently many other methods emerge
including Lindstedt-Poincare method [7,8], Krylov Bogoliubov averaging
Method (KBM) [9], Power-series method [10] which allow for the creation of
an approximation solution valid for large amplitudes of oscillation. Though
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these methods involve minimum algebraic manipulations, it is usually difficult to achieve higher-order analytic approximation to solutions. Recent
study reports based on higher order approximation [11-13] present the solution for nonlinear differential equation with odd and even nonlinearity that
immensely involves perplex calculations, computation, and consumption of
time. One among these is the Newtons’s Harmonic Balance Method (HBM)
[14-18]. An improvised version of standard Harmonic Balance Method is the
Linearized Harmonic Balancing approach [19,20] used for finding solution for
strong nonlinear differential equations very near to accuracy by more than
one approximation procedure. This method is significant over other methods in simplicity, non-requirement of initial condition from the outset and
numerical integration [4]. Also this approach holds good for small as well as
large amplitudes of oscillation.
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Duffing oscillators are extremely nonlinear that exhibit stunning behaviour
even if it is in unforced condition. Hopefully, no researches have done a
detailed dynamics about a system under a specific range of displacement
which is described by a Cubic Quintic Septic Duffing equation. This paper
presents the dynamics of three springs carrying equal stiffness constructed
geometrically with negative and positive stiffness elements which are kept at
a particular oblique angle of Quasi Zero Stiffness condition.
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The paper is systemised as follows: Details of the negative stiffness property of two springs and the configration of three-spring conservative system
for optimised QZS condition are given in Section 2. In section 3 we discuss
about the Force-Displacement phenomenon to analyse the system for the
displacement upto the maximum exertion range from equilibrium position
and arrive at second order nonlinear differential Duffing equation. Section
4 describes the Linearised Harmonic Balance Method (LHBM) to solve the
Duffing equation analytically upto third order approximation to observe the
natural frequency about the operating point. The exact solution by direct
integration of governing equation is included in Section 5. Section 6 provides
the frequency comparision results obtained from the proposed technique and
the direct integration method and further investigation of the prominent features of the Frequency-Amplitude graph. In addition, the behaviour of the
system with increase in time and the phase plot for a particular set of parameter values are examined . Section 7 compares the result of the governing
equation with the Cubic-Quintic Duffing equation. Finally, Section 8 details
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the brief results of the present work and ends with concluding comments and
future scope.
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2. Construction of three spring model
Two identical linear elastic springs each of length L0 having equal stiffness kh are joined at one end say B to hold the mass and other two ends are
separately hinged at two different supports A and C as shown in the Figure
1 (a). The springs are initially at an angle say θ0 which is at the height h0
from the horizontal.
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When a force is applied by placing a mass at B, the system behaves as
a negative stiffness mechanism. The free dynamics of this single degree of
freedom body is purely nonlinear when the mass reaches the horizontal position. This is because of the change of force in the springs exerted on the
mass along the vertical direction. This nonlinear force causes change in stiffness which results in the variation of vibration response in the system. For
instance, a sudden increase in dynamic stiffness causes reduction in vibration
resulting in high static stiffness which leads to undesirable deflection of mass.
This limitation can be succeeded by including another linear spring of stiffness kv parallel to the vertical component of the inclined springs as shown
in Figure 1 (b). The added vertical spring acts as a positive stiffness element.
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Thus the combined effect of negative and positive stiffness springs can
control the change in stiffness of the system. Hence, we obtain a smooth dynamics over a certain range of displacement around the horizontal position.
This phenomena of low dynamic stiffness around the equilibrium region is
known as Quasi Zero Stiffness (QZS). The QZS property allows the system
to respond with low natural frequency when it is disturbed from the equilibrium point. Also, this mechanism explicitly relies on the system parameters.
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However, there are infinite number of possible combinations of geometrical and stiffness system parameters to give QZS character in which the
optimised system parameter falls under the obligue angle ranging from 480
to 570 which would be determined from the relation cos θ0 = Ld00 [2]. To understand the free vibration of the three spring system under QZS condition,
we make an attempt to construct the system by a particular set of geometric
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Figure 1: (a) Arrangement of the Three-Spring system, (b) Three-spring system leading
to Cubic, Quintic and Septic oscillations
parameters say d0 =0.3m, L0 =0.5m which corresponds to the inclined angle
θ0 =530 .
3. Formulation of Duffing equation
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Let the system in Figure 1 is compressed to an angle say θ by placing an
arbitrary mass m at B so that the oblique springs (kh ) reaches the horizontal
position in such a way that it is at a height h from the equilibrium point. The
mass is supported rigidly upright by the vertical spring (kv ). Now disturb the
mass to some distance x from the horizontal position so that the resulting
force on the spring system is
F = 2kh ∆L sin θ + kv x,
(1)
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where ∆L = |L − Ld | is the change in length of the spring. L and Ld are the
length of the
p compressed and
p displaced springs respectively. They take the
form L = d20 + h2 , Ld = d20 + (h + x)2 and sin θ = h+x
.
Ld
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Hence Eq. (1) becomes
F = 2kh (h + x)[ √
L
d20 +(h+x)2
− 1] + kv x.
(2)
Binomial expansion of [d20 +(h+x)2 ]1/2 yields a polynomial equation with infinite number of terms raised to higher powers in which the terms upto seventh
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power is considered here to describe the dynamics of the three-spring system.
Now Eq. (2) is written as
Lkh
Lkh
(h + x) − 2kh (h + x) − 3 (h + x)3 +
d0
d0
3 Lkh
15 Lkh
(h + x)5 −
(h + x)7 + kv x.
5
7
4 d0
24 d0
2
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F =
(3)
F =
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The system is stable for QZS condition at the equilibrium point, h = 0. Thus
the above equation is reformed into
kh 3 3 kh 5 15 kh 7
x +
x −
x + kv x.
d20
4 d40
24 d60
(4)
The equation of motion of the system having equal stiffness (kh = kv = k)
for all the three springs is obtained from F − (−mẍ) = 0 as
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d2 x
+ αx(t) − βx3 (t) + γx5 (t) − δx7 (t) = 0,
dt2
15 k
24 md60
are
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3 k
k
k
where α = m
is the linear coefficient and β = md
2 , γ = 4 md4 , δ =
0
0
the cubic, quintic, septic power coefficients respectively.
(5)
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This equation is a second order ordinary differential equation of mass
holding odd nonlinear forces and is termed as Cubic Quintic septic conservative Duffing equation. This equation normally gives the real dynamics of the
model undergoing moderately with small as well as large amplitudes of motion. But, our interest is to find the system dynamics about the equilibrium
position where the dynamic stiffness is found to be low that results in low
natural frequency. For this, we try to calculate numerically the change in
frequency of the system for different initial displacements by assigning a set
of values for the geometric parameters k = 200N/m, d0 = 0.3m, m = 2.8kg,
L = 0.5m.
Generally, the low stiffness region lies in a certain displacement range
about the equilibrium point. In this region, the dynamic stiffness is zero at
the equilibrium point which corresponds to low natural frequency and near
the neighbourhood of the equilibrium point for short length the stiffness of
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the
the
the
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the system is low. The expressions for the static equilibrium point and
maximum excursion distance as a function of maximum stiffness from
equilibrium point are calculated from the stiffness equation explored in
Force-Displacement characteristic theory [2] and are given as
√
xeq = L0 1 − d2 ,
q
dp = L0 d [1 − K(1 − d)]−2/3 − 1,
(6)
(7)
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where d = Ld00 . It is determined that the value of d for which dp is a maximum, changes depending on the value of the prescribed stiffness K. It is
clear that the above two relations (6) and (7) only varies with the geometrical
parameters.
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For the chosen set of geometrical parameters the above expressions yield
the values for the equilibrium point xeq and the maximum exertion distance
dp as xeq = 0.4, dp = 0.12 for K = 0.5. By using these values we can define
a range xeq − dp < x < xeq + dp , that is from x = 0.28 to 0.52, where the dynamic stiffness is found to be low and the overall performance of the system
under this range leads to negative stiffness mechanism.
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It is possible to determine the abstract ideas that have mentioned above
from the solution of the equation of motion of the three-spring sytem. To
examine this we substitute the values of k, d0 , m in α, β, γ, δ to obtain
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d2 x
+ 71.43x − 793.65x3 + 6613.75x5 − 30619.24x7 = 0.
(8)
2
dt
This Duffing equation involves large system parameters with strong fifth
and seventh power nonlinearities. Though Eq. (8) looks simple in apperance,
it is rather difficult to solve exactly due to the presence of Cubic, Quintic
and Septic nonlinear force terms. However, for large nonlinear parameters,
the Higher order Approximation Method finds suitable to provide more accurate higher-order approximate frequencies and the corresponding periodic
solutions for various initial amplitudes compared with the exact solution.
4. Proposed Analytical Methodology
For the higher order approximation technique, linearisation of governing
equation is performed earlier to the Harmonic Balance Method to estabilish
approximate expressions for frequencies and periodic displacements.
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4.1. Linerization of governing equation
The presence of nonlinear restoring force function of x in Eq. (8) limits the occurance of a periodic solution in t. By the action of linearization,
one can assume a periodic solution satisfying the initial condition x(0) =
(0) = 0.
A; dx
dt
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This is done by introducing a new independent variable τ = ωt such that
the solution of Eq. (8)
i) is a periodic function of τ
ii) satisfies the above initial condition and
iii) whose amplitude A depend on frequency.
Thus Eq (5) is rewritten in terms of angular frequency ω as
ω2
d2 x(τ )
+ αx(τ ) − βx3 (τ ) + γx5 (τ ) − δx7 (τ ) = 0.
2
dt
(9)
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Expanding Eq. (9) with respect to small increments ∆x1 (τ ) and ∆ω12 at
x(τ ) = x1 (τ ) and neglecting the powers of x1 greater than seven and linearizing with respect to x1 (τ ) yields
2
d2 x1
2 d x1
+
ω
∆
+ αx1 − βx31 + γx51 − δx71 +
dt2
dt2
(α − 3βx21 + 5γx41 − 7δx61 )∆x1 = 0.
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ω2
(10)
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Let x1 (τ ) = A cos τ is a periodic function of period 2π which is considered
as the initial approximation to x(τ ). Simplifing Eq. (10) gives
d 2 x1
+ αA cos τ + α∆x1 − βA3 cos3 τ
dt2
−3βA2 cos2 τ ∆x1 + γA5 cos5 τ + 5γA4 cos4 ∆x1 − δA7
cos7 τ − 7δA6 cos6 τ ∆x1 = 0.
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−ω12 A cos τ + ω12 ∆
(11)
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This is a linear equation in ∆x1 and also a periodic function of τ of period
2π.
4.2. Harmonic Balancing Method
After linearization, Newton’s Harmonic Balance procedure is used to solve
for approximate angular frequency and periodic displacement. Before applying this procedure the above linear Eq. (11) should undergo the Fourier series
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expansion of odd and even powers of cos τ .
The Fourier coefficients for Eq. (11) are found to be
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3
5
35
a1 = αA − βA3 + γA5 − δA7 ,
4
8
64
1 3
5
21
a3 = − βA + γA5 − γA7 ,
4
16
64
1
7
1
a5 = γA5 − δA7 , a7 = − δA7
16
64
64
15 4 35 6
γA − δA ,
4
8
3 2 5 4 105 6
b2 = − βA + γA −
δA ,
2
2
32
5
21
7
b4 = γA4 − δA6 , b6 = − δA6 ,
8
16
32
b8 = 0.
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b0 = 2α − 3βA2 +
(12)
(13)
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4.2.1. First order approximation
For the first order approximation, we set ∆x1 (τ ) = 0, and ∆ω12 = 0.
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The initial approximation to the periodic function becomes x(τ ) = x1 (τ ) =
Acosτ. Substituting Eqs. (12), (13) and the above approximation in Eq. (11)
gives
−Aω12 cos τ + a1 cos τ + a3 cos 3τ + a5 cos 5τ + a7 cos 7τ = 0.
(14)
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Setting the coefficient of cos τ to zero consequences first order analytical
approximate frequency ω1 and periodic solution x1 (t) as
q
35
δA6 ,
ω1 (A) = α − 43 βA2 + 58 γA4 − 64
x1 (t) = A cos(ω1 t).
(15)
Here the angular frequency ω1 as well as the periodic solution x1 (t) depend
on the initial amplitude A.
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4.2.2. Second order approximation
To construct second order analytical approximate solution, the correction
term ∆x1 is extended to ∆x1 (τ ) = k1 (cos τ − cos 3τ ) and ω2 , x2 takes the
form ω2 = ω1 + ∆ω1 , x2 (τ ) = x1 (τ ) + ∆x1 (τ ).
Substituting the above correction term and Eqs. (12), (13) in Eq. (11)
and applying trignometric series expansion, we obtain
−(ω12 + ∆ω12 )A cos τ + ω12 ∆
d2 x1
+ (a1 cos τ + a3 cos 3τ
dt2
b0
+ b2 cos 2τ + b4 cos 4τ +
2
b6 cos 6τ + b8 cos 8τ )∆x1 = 0.
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+a5 cos 5τ + a7 cos 7τ ) +
(16)
Setting zero coefficients to cos τ and cos 3τ results in the production of
quadratic equations which on solving simultaneously yields the expression
for k1 and ∆ω12 as
2Aa3
,
−18a1 + a(b0 − b2 − b4 + b6 )
a3 (−2a1 + (b0 − b4 )A)
∆ω12 =
.
−18Aa1 + A2 (b0 − b2 − b4 + b6 )
(17)
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k1 =
The second order approximated angular frequency ω2 is solved as
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ω2 =
3
5
35
[α − βA2 + γA4 − δA6 ]1/2
4
8
64
a3 [−2a1 + (b0 − b4 )A]
+[
]1/2 .
2
−18Aa1 + A (b0 − b2 − b4 + b6 )
(18)
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It has more accuracy than ω1 which is acquired from first order approximation
technique. The corresponding approximate periodic solution x2 (t) is obtained
as
x2 (t) = a cos(ω1 t) + k1 [cos(ω2 t) − cos 3(ω2 t)].
(19)
Here also, both the frequency and periodic solution of second order approximation depends on the amplitude A.
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4.2.3. Third order approximation
For the third order analytical approximation, the terms x1 (τ ) and ∆x1 (τ )
are replaced by x2 (τ ) and ∆x2 (τ ) respectively.
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The approximation is set as ∆x2 (τ ) = k2 (cos τ − cos 3τ ) + k3 (cos 3τ −
cos 5τ ) and ω3 , x3 are assigned as ω3 = ω2 + ∆ω2 , x3 (τ ) = x2 (τ ) + ∆x2 (τ ).
Substituting Eqs. (12), (13) in Eq. (11) and setting the coefficients of
cos τ , cos 3τ and cos 5τ to zero in the trignometric series expansion gives the
following expressions for k2 , k3 and ∆ω32 :
(20)
k3 =
p8 (−p2 p9 + p1 p10 ) + p4 (p6 p9 − p5 p10 )
,
p8 (−p3 p10 + p2 p11 ) + p4 (p7 p10 − p6 p11 )
(21)
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p8 (p3 p9 − p1 p11 ) + p4 (−p7 p9 + p5 p11 )
,
p8 (−p3 p10 + p2 p11 ) + p4 (p7 p10 − p6 p11 )
p3 (−p6 p9 + p5 p10 ) + p2 (−p7 p10 + p6 p11 ) + p2 (p7 p9 − p5 p11 )
,
p8 (−p3 p10 + p2 p11 ) + p4 (p7 p10 − p6 p11 )
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∆ω22 =
k2 =
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where
(22)
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p1 = −2Aω22 + 2c1 − 2k1 ω22 , p2 = d0 − d4 − 2ω22 ,
p3 = d2 − d6 , p4 = −(2A + 2k1 ), p5 = 9k1 ω22 + c3 ,
d0 d6
d0 d2 d6
+
− , p7 = −9ω22 +
+
p6 = 9ω22 −
2
2
2
2
2
d2 d8
1
− − , p8 = 9k1 , p9 = c5 , p10 = (d4 + d6 − d2
2
2
2
1
−d8 ), p11 = (d2 + d8 − d10 − d0 ) + 25ω22 .
2
(23)
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Eqs. (20), (21), (22) are solved for Fourier coefficients c1 , c3 , c5 , c7 , d0 , d2 ,
d4 , d6 , d8 , d10 and the periodic solution x3 (t) and its corresponding angular
frequency ω3 are found to be
x3 (t) = (A + k1 + k2 ) cos ω32 t + (k3 − k1 − k2 )
cos 3(ω32 t) − k3 cos 5(ω32 t),
ω32 = ω22 + ∆ω22 .
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5. Direct Integration for Accurate Solution
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Angular frequency ω3 derived from the third order approximation is more
accurate than the angular frequencies ω1 , ω2 obtained from first and second
approximation methods. This accuracy is noticed graphically in the Amplitude - Frequency plot shown in Figure 2.
The governing Cubic Quintic Septic Duffing equation (5) is integrated
directly under the initial condition x(0) = A and ẋ(0) = 0 to obtain the
exact frequency ωex of the dynamical three-spring system.
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Eq. (5) is integrated once and applying the above initial condition gives
rise to
1 2 1 4 1 6 1 8
αA − βA + γA − δA = C.
2
4
6
8
(25)
0
A
dx
q
. (26)
α(A2 − x2 ) − β2 (A4 − x4 ) + γ3 (A6 − x6 ) − 4δ (A8 − x8 )
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T (A) =
Z
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where C is the integration constant.
One can find the solution for the period of oscillation of the governing
equation by equating Eq. (25) and Eq. (5) after integration. The result is
PT
By substituting x = −A cos t in Eq. (26), we get
T (A) =
Z
π
2
0
dt
√
R
(27)
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where R = α − βA2 − 21 δA6 − 12 δA6 cos4 t + 12 βA2 sin2 t + 14 δA6 sin2 t +
1
δA6 sin2 cos2 t.
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The exact frequency which is a function of initial amplitude A is obtained
from the relation ωex (A) = T2π
and the exact displacement is expressed as
(A)
xex (t) = A cos t where
Z
dx
q
t=
.
(28)
α(A2 − x2 ) − β2 (A4 − x4 ) + γ3 (A6 − x6 ) − 4δ (A8 − x8 )
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6. Frequency Comparison
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The three angular frequencies derived from the approximation method
in this paper are denoted as ω1 , ω2 , ω3 respectively. The exact frequency is
denoted by ωex . The frequencies calculated by the approximation method is
compared with the exact frequency which is obtained by the integration of
governing Duffing equation and is shown in Figure 2.
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Figure 2 depicts the frequency response of free vibration of the threespring system under QZS condition. The change in angular frequency determined using the LHBM method for various initial displacements shows that
the curve for ω3 almost coalesce with the curve for ωex whereas curves ω1
and ω2 start to deviate from ωex at x = 0.36. This indicates that the third
order approximate frequency ω3 provides excellent approximation with the
exact frequency ωex as compared to ω1 and ω2 . The natural frequency of the
system is observed from the figure and it is found as ωn = 8.44N m−1 Kg −1 .
Another noticeable observation extracted from the figure is the value of the
equilibrium point, which falls at xeq = 0.41. This value is exactly close to
the expected value xeq = 0.4 determined from Eq (6). Also, the angular
frequency is very low at the equilibrium point due to zero dynamic stiffness
and increases when the system is away from the equilibrium point. The displacement upto which the angular frequency is found to be low from either
sides of the equilibrium point is observed as x = 0.12 which corresponds to
the displacement range 0.29 < x < 0.53 where the natural frequency and
stiffness are minimum near the equilibrium point. This measurement virtually matches with the calculated range 0.28 < x < 0.52. This means that
outside this typical range the dynamic stiffness increases leading to higher
angular frequency.
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More generally, the natural frequency of the spring mass system depends
only on the material properties called stiffness and mass. Normally, a stiffer
spring or a lower mass increases the natural frequency and a softer spring or
a higher mass lowers the natural frequency. For increasing spring stiffness
keeping mass as constant, we find that the natural frequency (ωn ) of the
system increases and is shown in Figure 3 (a). Likewise the decrease in
natural frequency with increasing mass with constant stiffness is presented
in Figure 3(b).
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Figure 2: Comparison of calculated frequencies with exact frequency of Cubic Quintic
Septic Duffing Equation for α = 71.43, β = −793.65, γ = 6613.75, δ = −30619.24
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Figure 3: (a) Increase in Natural frequency with increasing Stiffness, (b) Decrease in
Natural frequency with increasing Mass
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The periodic solution for the Cubic Quintic Septic Duffing equation for
different amplitudes within the range x = 0.28 to x = 0.52 are plotted in
Figure 4. These figures represent that the period of oscillation is not constant all over the range and it depends on the initial displacement of the
mass in the system about the equilibrium point. As the distortion amplitude
become shorter which means the amplitude near the neighbourhood of the
equilibrium point, between x = 0.4 and x = 0.43 the period of oscillation
increases, therefore the angular frequency decreases. Beyond x = 0.43 there
is a change in period of oscillation due to the high dynamic stiffness response
in the structure.
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The corresponding phase diagrams are also given in Figure 5 which shows
an elongated perfect ellipse at the low frequency region indicating smooth
dynamics with respect to initial amplitudes near the equilibrium point. The
perfectness of the ellipse gets distorted along the vertical axis for the displacement of mass far from the equilibrium point xeq = 0.41. This is because of
the oscillation characteristic of Cubic, Quintic, Septic nonlinear force springs.
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Figure 4: Approximate Periodic Solution of Cubic Quintic Septic Duffing Equation with
α = 71.43, β = −793.65, γ = 6613.75, δ = −30619.24 for the initial displacement range
X = 0.1 to 0.51
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Figure 5: Phase diagrams of Cubic Quintic Septic Duffing Equation with α = 71.43, β =
−793.65, γ = 6613.75, δ = −30619.24 for the initial displacement range X = 0.1 to 0.51
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Figure 6: (a) Frequency response of Cubic Quintic Duffing equation, (b) Phase diagram
of Cubic Quintic Duffing equation, (c) Phase diagram of Cubic Quintic Septic Duffing
equation
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7. Comparison of result with Cubic Quintic Duffing oscillator
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The behaviour of the solution x(t) changes for the parameter values α, β,
γ and δ. If the system is modeled for the Duffing equation representing the
Cubic and Quintic restoring force terms then one cannot expect the same
situation discussed above as for the Cubic Quintic Septic form under same
α, β, γ values. Because variation in the strength of restoring forces in the
system results into variety of interesting dynamics. When δ = 0, the system
turns into Cubic Quintic Duffing oscillator. Eq. (5) is written as
d2 x(t)
dt2
+ αx(t) − βx3 (t) + γx5 (t) = 0.
The Frequency - Amplitude curve of the above equation for the same
system parameters is shown in the Figure 6(a). The disparity in the dynamical behaviour corresponds to the number of equilibrium points in the system.
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Mostly, an unforced Cubic Quintic Duffing equation with appropriate system
parameters can produce one, three, and five equilibrium points. Similarly a
Cubic Quintic Septic Duffing equation can generate upto seven equilibrium
points. In the case of the three spring system describing Cubic Quintic Duffing equation we observe only one equilibrium point at the center (0, 0) where
as the Cubic Quintic Septic Duffing equation yields three equilibrium points
one at the center (0, 0) and other two are at (±0.4124, 0). These are reflected
in the phase portraits shown in Figures 6 (b) and 6 (c) respectively. But,
the system parameters have no effect on the natural frequency of the system
because the material property known as stiffness and mass of the system
determine the natural frequency. In both the Duffing equations we considered same stiffness and mass. Hence, both the systems have same natural
frequency as ωn = 8.44N m−1 Kg −1 .
8. Conclusion
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In the present work, a nonlinear model is constructed using three springs
one of which act as positive stiffness and other two springs provide a negative
stiffness with the geometrical parameters d0 = 0.3, L0 = 0.5, θ0 = 530 satisfying the QZS property at the equilibrium point. The Force-Displacement
characteristics of this spring configration leads to a strong nonlinear conservative Cubic Quintic Septic Duffing equation consisting of large nonlinear
restoring force terms. An analytical higher order approximation technique
based on the combination of linearization of governing equation with the Harmonic Balance method (LHBM) is used to solve the equation. This approach
enables us to estabilish approximate expressions for angular frequency and
periodic solution for different initial amplitudes about the equilibrium point.
The present method provides an excellent accuracy of the approximate frequency with the exact frequency. Using this technique we observe the natural
frequency of the system to be ωn = 8.44N m−1 Kg −1 and it is minimum at
the equilibrium point. The change in natural frequency for varying spring
stiffness and mass is also discussed here. Furthermore, a certain displacement
range is noticed as 0.29 < x < 0.53 where the dynamic stiffness is found to
be low corresponds to low natural frequency near the equilibrium. The periodic solution and phase plots of the three-spring system for various initial
amplitudes are analysed based on the stability of the system. In addition,
the results are compared with the Cubic Quintic Duffing equation in which a
disparity in dynamical behaviour is found due to the variation in the number
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of equilibrium points in the system.
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In practice, numerous applications make use of Duffing oscillators are
performed by powering the system with suitable excitation force. The reason
is that the Duffing oscillators are more sensitive to both the friction and
driving force. So, our future work is in process that referring the system to
an externally excited Duffing oscillator which adds another dimension to the
system.
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