Ijariie1165

Vol-1 Issue-2 2015 IJARIIE-ISSN(O)-2395-4396
1165 www.ijariie.com 240
Study of Non-Linear Behavior of Vibrating
System
Chetan S.Dhamak1
, Dipak S.Bajaj2
, Vishnu S.Aher3
, Kapil K. Dighe4
1,2,3
Mechanical Engineering, Amrutvahini College of Engineering, Sangamner, Maharashtra, India
ABSTRACT
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. In the case of the
real structures a linear model will be insufficient to describe the dynamic behavior correctly. It thus appears natural
to introduce non-linear models of structures which are able to predict the dynamic behavior of the real structures.
This seminar includes study of non-linear vibrations, it different types and various applications. Here the vibratory
behavior of an absorber of vibration related to a system subjected to a harmonic load, in the presence of
uncertainties on the design parameters is optimised. The total system is modeled by two degrees of freedom (2 dof)
with a shock absorber and a generalized non-linear stiffness. one proposes to optimize the vibratory behavior of an
absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the
design parameters. The total system is modelled by two degrees of freedom (2 dof) with a shock absorber and a
generalized non-linear stiffness. The resolution is carried out in the temporal field according to a traditional
diagram. It is a question of seeking the optimal responses envelopes of the deterministic and stochastic case and this
for the non-linear displacements, phases and forces.
Keyword: - Non-Linear, Oscillation, Absorber and Vibration
1. INTRODUCTION
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. In order to
reduce the vibrations in the revolving machines and the mechanical systems, the dynamic absorbers are often used in
various mechanical applications (Crankshaft, rotor of the wings of a helicopter, etc.). In the case of the real
structures a linear model will be insufficient to describe the dynamic behavior correctly. It thus appears natural to
introduce non-linear models of structures which are able to predict the dynamic behaviour of the real structures. The
solutions of these non-linear problems are obtained by approximate methods which exploit iterative algorithms.
In this paper, one proposes to optimize the vibratory behaviour of an absorber of vibration related to a system
subjected to a harmonic load, in the presence of uncertainties on the design parameters. The total system is modeled
by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. The resolution is
carried out in the temporal field according to a traditional diagram. It is a question of seeking the optimal responses
envelopes of the deterministic and stochastic case and this for the non-linear displacements, phases and forces.
The multi-objective optimization step consists in seeking the first Pareto front of several linear and nonlinear
objective functions by using a genetic algorithm of type “NSGA” (Non-dominated Sorting Genetic Algorithm). The
design parameters are: mass, linear and non-linear stiffness and damping of the absorber. To obtain solutions
presenting a good compromise between optimality and the robustness, one introduces uncertainties on these design
parameters. The robustness is then defined by the dispersion of the parameters (definite as the ratio: mean
value/standard deviation) and it is introduced as additional objective function.
The use of the clusters resulting from the Self-Organizing Maps of Kohonen (SOM) is also suggested for a rational
management of the design space. A study of sensitivity a posteriori can be exploited in order to eliminate the non-
significant design parameters.[1]

1.1 Non Linear Vibrations
Most physical systems can be represented by liner differential equation. A general equation of this type is
.In this equation which is for linear system, the inertia force, the damping force and the
spring force are linear functions of respectively. This is not so in the case of non-linear systems. A
general equation on non-linear system is in which the damping force and the spring
force are not linear functions of . There are quite some physical systems which have non-linear spring and
damping characteristic. Rubber springs and other similar isolators have spring stiffness which increases with
amplitude. Cast iron and concrete have spring stiffness which decreases with amplitude. Examples of non-linear
damping are dry friction damping and material damping. Even so called linear systems tend to become non-linear
with larger amplitudes of vibration. The analysis of non-linear systems is difficult. In some cases there is no exact
solution.[1]
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1.2 Application
Linear vibration isolators are only useful if their natural frequencies are well below the excitation frequency. Thus,
they are limited to such applications as moderate environmental disturbances. However, under severe environmental
disturbances such as shocks, impact loads, or random ground motion, their spectrum will definitely contain
dangerous low-frequency components. The isolator under these conditions experiences excessive deflections that
can cause over-stress and even damage to the system. For this reason, it is imperative to consider effective nonlinear
isolators, which can serve several applications, such as
1. Reducing line spectra in the radiated acoustical signature of marine vessels.
2. Isolating equipment mounted in ships navigating in extreme sea waves.
3. Reducing the magnitude of the high launch loads across all frequency bands acting on spacecraft.
4. Reducing severe vibrations due to impact loads.
5. Protecting buildings, bridges, liquid storage tanks, oil pipelines, and nuclear reactor plants against the damaging
effects of earthquakes.
6. Isolating laser interferometers of gravitational wave detectors.
7. Isolating electronic equipment, automotive vehicle front-end-cooling systems, and passengers from road
roughness excitation.
8. Isolating automotive power-train system, engine through proper design of rubber and hydraulic mounting
systems.
9. Protecting operators of hand-held machines.[4]
1.3 Types of Non-Linear Vibration Isolators
1.3.1 Ultra-Low-Frequency Vibration Isolators
The vibration isolation of mirrors in laser interferometers used in gravity wave detection is considered an important
factor in the success of the Laser Interferometer Gravitational Wave Observatory (LIGO) and VIRGO projects.The
sensitivity of terrestrial gravitational detectors is limited at low frequencies by the resulting degree of seismic
isolation. Efforts to improve this isolation have resulted in what is known as ultra-low-frequency isolators.
Physicists introduced what is called a pre-isolator, which is in principle an isolation stage, which is designed to have
a very low resonant frequency of suspension. The main distinction between a pre-isolator and normalisolator is that
structures that are designed to have very low resonant frequencies tend to be larger and more massive, and thus
possess lower natural frequencies than normal isolator chain stages. As a result they generally do not provide useful
isolation in the 10 Hz–1 kHz detection band but are included mainly to reduce residual motion at frequencies below
the detection band. The worst residual motion usually occurs close to 0.5–1 Hz at resonant frequencies of the lowest
swinging modes of a standard isolation chain. Good design of a pre-isolation system can easily reduce the seismic
drive to these modes and thus the residual low-frequency motion by two orders of magnitude.

Fig.1 Force and base motion nonlinear isolators: (a) Watt’s linkage force isolator, (b) inverted pendulum retrained
by a flat short spring, and (c) two opposite pendulums.
Typical examples of ultra-low-frequency horizontal isolators are shown in Fig.1.shows the Watt’s linkage with a
mass suspended from an appropriate point which moves along a circle of very large radius. The inverted pendulum
shown in Fig. is restrained by a short flat spring to provide positive restoring moment.[4]
1.3.2 Euler Column Isolators
An Euler spring is a column of spring steel material that is compressed elastically beyond its buckling load. The
analysis of finite deflections of prismatic elastic columns after buckling was developed by Euler using elliptic
integrals. The shape of the elastic curve in the post-buckling state is referred to as the elastic.[4]
1.3.3 Gospodnetic-Frisch-Fay Beam Isolator
The static deflection curve of a thin elastic beam forced to deform by three symmetrical frictionless knife edged
supports, was analyzed by Gospodnetic and documented by Frisch-Fay. Since the beam is inextensible, there is no
limitation on its deflection, and a closed form solution for the deflection curve was given in terms of elliptic
functions. Fig. 2 shows a schematic diagram of a beam similar to the Gospodnetic–flexible beam. It is free to slide
on two knife-edged supports under the action of the load P. This beam can be used as a resilient isolator between the
machinery and the base in marine vessels. The beam can also model a load carrying bearing for pressure pipelines
against earthquake ground motion [4]
Fig.2 Schematic diagram of deflected under static load of a flexible beam and free to slide at two knife edge
supports.

1.3.4 Nonlinear viscoelastic and composite isolators
Viscoelastic materials are widely used for vibration isolation in automotive and aerospace industries. It is well
known that the shear modulus and loss factor of rubber materials depend on frequency, ω, as well as temperature, T.
Thus one can express the complex shear modulus, G* (ω, T), in the form G* (ω, T) = G’ (ω, T) + iG” (ω, T) = G’
(ω, T) [1 + iη (ω, T)], Where G’ (ω, T) is the shear modulus and G’ (ω, T)η (ω, T) is the loss factor.The dependence
of these two parameters on the temperature and frequency is shown in Fig. 3. shows the optimum regions of
viscoelastic materials for different types of damping devices. For example, region A is optimum for free-layer
treatments, having a high modulus and high loss factor. On the other hand, region B is optimum for constrained-
layer treatments, possessing a low modulus and high loss factor. Region C is optimum for tuned-mass dampers
possessing a low modulus and low loss factor.
Fig. 3 Dependence of shear modulus and loss factor on temperature at constant frequency/or frequency at constant
temperature.
With reference to Fig. 3 one can see that at the so-called rubber-to-glass transition, the loss factor passes through a
maximum value that lies approximately in the frequency or temperature range through which the shear modulus
changes rapidly. The frequency at which the shear modulus increases rapidly as the excitation frequency increases is
called the transition frequency. On the other hand, the temperature at which the shear modulus decreases rapidly as
the temperature increases is called the glass transition temperature. The glass transition state is a non-equilibrium
state at which the nonlinear viscoelastic nature of structural recovery can lead to surprising behaviour. Thus, one can
use the temperature and frequency as control parameters to study such phenomena as bifurcation and other complex
dynamic characteristics.[4]
1.4 Optimization Technique
In optimization studies including multi-objective optimization, the main focus is placed in finding the global
optimum or global Pareto-optimal frontier, producing the best possible objective values. However in practice, users
may not always be interested in finding the global best solutions, particularly if these solutions are quite sensitive to
the variable perturbation which cannot be avoided in practice. In such cases practitioners are interested in finding the
so-called robust solutions which are less sensitive to variables. Although robust optimization has been dealt in detail
in single-objective optimization studies, in this paper, we present two different robust multi-objective optimization
procedures, where the emphasis is to find the robust optimal frontier instead of global Pareto-optimal front.

2. MECHANICAL SYSTEM
Construction of Mechanical System
The studied system comprises:
 A metal bar of free embedded constant section (principal system)
 Two identical wings, each one is embedded with the bar of with dimensions and is dependent on a mass of the
other with dimensions. Each wing is made of two blades of constant section. The unit, thermal switches with
masses, constitutes the dynamic absorber (secondary system).
Fig.4 Mechanical system
Each system (principal and secondary) is brought back to only one dof. The principal system is characterized by a
mass m1, a linear spring of stiffness k1, a linear shock absorber c1, two localized non-linearities of stiffness
respectively k1
’
of variable power and cubic k1
”
and the same for damping with the two respective coefficients of
non-linearities c1
’
and c1
”
. It is subjected to a harmonic load of variable amplitude and pulsation ω, m2,k2, c2,k2
’
, k2
”
,
c2
”
and c2
’
. Fig.5 represents the total system.
Fig.5 Basic Model

The equations of motion of the system can be written as follows:
For the continuation, one calls non-linear force on a mass mi the sum of these two efforts. The positive real
parameter r is introduced in order to consider various situations of non-linearities to knowing:
r = 3: Only cubic non-linearity;
r = 2: Quadratic and cubic non-linearities;
r = 1.5: Non-integer and cubic non-linearities.
The last cases constitute what one will call generalized nonlinearity. Eqs. (1) are reduced to the matrix form:
[M]{ } + [Cnl]{ } + [Knl]{u} = {F} - - - - - - - - - - - - (2)
[M], [Knl] and [Cnl] are respectively the masses, non-linear stiffness and non-linear damping matrix. These matrixes
are written as follows:
Thereafter, one uses a dimensional writing by carrying out the following changes of variables:
After division by (m1xsta), the equation of motion (2) becomes:
[ ] { } + [Cnl( )] { } + [Knl( )] { } = { } - - - - - - - - - - - - - - - - (5)

For the solution of these equations, one uses the non-linear terms of the principal system (k1
’
,k1
”
, 1
’
, 1
”
) non-linear
terms of the secondary system (k2
’
, k2
”
, 2
’
, 2
”
) therefore that basic terms 2, xsta,ω1 and 2.The resolution is carried
out in the temporal field according to a traditional scheme of Newmark with linear acceleration in order to find the
temporal response of the system with absorber subjected to a harmonic load.[1]
3. EXPERIMENTAL ANALYSIS
The importance of the methodology that we have along with its performances is highlighted by two numerical
examples. Initially the mechanical behaviour of the total system through the representation of the curves: X1(t)
(displacement of the principal system) and X2(t) (displacement of the absorber) is illustrated. These curves are given
for three types of generalized nonlinearities (r = 2, r = 3 et r = 1.5). One will take in what follows 2= 0.01, xsta=
0.05 m, ω1= 70 rad/s, t = 3×10−4 s, 2= 0.01 and α1= α2= 0.001.
In the second time, one is interested in optimization of the absorber 2, k2
’
, k2
”
and 2 by seeking its linear and non-
linear optimal characteristics.
3.1Non-Linear Responses And Generalized Deterministic Diagrams of Phase
3.1.1 r = 2; Case: Quadratic Non-Linear Damping And Stiffness
The mechanical characteristics of the system are: k1
’
= k2
’
= c1
’
= c2
’
=5 and
k1
”
= k2
”
= c1
”
= c2
”
=0.25
the evolution of note displacements that the damper correctly absorbs the vibration in this configuration. From a
certain time “t” closes to 0.26 adimensional of both dof according to time. The oscillations of the absorber in their
turn are attenuated by a beat phenomenon

Fig.6 Standardized temporal non-linear responses (r = 2: non-linear damping and
stiffness).
Fig. 7 (a)
Fig. 7(a) illustrates the diagrams of phase for the principal system and the absorber. It’s noted that the absorber has
an elliptic orbit whose hearth coincides with the attractor one of the principal system.
Fig. 7 (b)
Fig. 7(b) illustrates the phase diagram of the main system and shows the existence of two very close attractors which
are clarified

Fig. 7 (c)
Fig. 7 (c) illustrates the phase diagram of the absorber and also shows the existence of two very close attractors.
Fig. 7 Diagrams of phase (r = 2: non-linear damping and stiffness).
3.1.2 r = 3; Case: Cubic Non-Linear Damping And Stiffness
’
= k2
’
= c1
’
= c2
’
=5 and k1
”
= k2
”
= c1
”
= c2
”
=0.25
Fig. 8 (a) Displacements in time
Fig.8 (b) diagrams of phase

Fig. illustrates the phase diagram of the main system. Here, two attractors are more visibly than for the quadratic
non-linear damping and stiffness case. This notifies a predominance of the cubic case compared to the quadratic
case.
Fig.8 (c) Trajectory
Fig. illustrates a complex trajectory of the global system in the plane (X1, X2), then an instable motion
is clearly notable.
3.1.3 r = 1.5; Case: Non-Integer Power Non-Linearity Combined With Cubic Non- Linearity Damping And Stiffness
’
= k2
’
= c1
’
= c2
’
=5 and k1
”
= k2
”
= c1
”
= c2
”
=0.25
Fig. shows the displacements evolution for the two dof according to time. It is noted that the auxiliary system
completely absorbs the vibration in this proportioning configuration. It is as to note as the movement is relatively
irregular similar of specified non-linearity. That may be one of the effects of the generalized non-linearity for the
non-integer power less than 2.
Fig. 9.r = 1.5 Standardized Temporal Non-Linear Responses.
Fig. illustrates the phase curves of the main system and the absorber. An existence of two attractors very brought
closer with an oval cycle orbit is notified.

Fig. 10.Diagram of phase (r = 1.5: non-integer non-linear damping and stiffness).
Figs. 11(a) and 11(b) illustrate the non-linear applied forces on each mass according to the corresponding
displacement. A dominant parabolic form is noted for the two dof, a fast convergence towards the steady balance
point is visible. These results, compared with the generalized quadratic case, give a significant difference especially
for the absorber.
(a) (b)
Fig. 11. (A) And (B) Non-Linear Forces (R = 1.5: Non-Integer Non-Linear Damping And Stiffness).
Fig. 12 illustrates the global system complex trajectory in the plane (X1, X2).
Fig. 12.Trajectory (r = 1.5: non-integer non-linear damping and stiffness).

4. CONCLUSION
In this work, we proposed a mechanical system model led by only one dof relating to its first mode and equipped
with a dynamic absorber with only one dof too.Thus the system obtained comprises a cubic and quadratic
generalized non-linearity combined stiffness and of damping. We used the numerical diagram of Newmark with
linear acceleration in order to find the temporal response of the principal system with absorber subjected to a
harmonic load.Deterministic calculations made it possible to highlight the contribution of this type of non-linearity
on the absorption of the vibrations and the behaviour of the non-linear system through the curves of phases
(attractile, ovalization, etc.).
The presentation of the bifurcation diagram of the absorber in the cubic case for the chosen parameters illustrates
three kinds of motions in several frequency bands: periodic, quasiperiod and chaotic motions. The phenomenon of
bifurcation appears clearly in this diagram.In the optimization part, one determined the optimal characteristics of the
damper located on the first face of Pareto with the compromise optimality-robustness. These solutions contribute to
the robust design of the non-linear system.
For the frequency (f = 9.8 Hz) corresponding to the chaotic motions, one robust optimized solution is selected and
analyzed. The bifurcation diagram obtained for this solution shows a particular move from the chaotic motion to
quasiperiodic motion. That means that, after robust optimization the motion may change of nature and then as
regards the system parameters different motions can be obtained.
REFERENCES
[1] M.L. Bouazizi, S. Ghanmi, R. Nasri, N. Bouhaddi,10 April 2008, “Robust Optimization of The Non-
Linear Behaviour of A Vibrating System”,European Journal of Mechanics A/Solids 28 (2009) 141–154
[2] Yanqing Liu, Jianwu Zhang, 25 July 2001, “Nonlinear Dynamic Responses of Twin-Tube Hydraulic Shock
Absorber”,Mechanics Research Communications 29 (2002) 359–365
[3] Keith Worden, Daryl Hickey, MuhammedHaroon, Douglas E.Adams, 27 March 2007, “Nonlinear System
Identification of Automotive Dampers: A Time And Frequency-Domain Analysis”,Mechanical Systems
and Signal Processing 23 (2009) 104–126
[4] R.A. Ibrahim, 4 January 2008, “Recent Advances In Nonlinear Passive Vibration Isolators” Journal of
Sound and Vibration 314 (2008) 371–452
[5] A.K. Samantaray, 15 March 2007 “Modeling And Analysis Of Preloaded Liquid Spring/ Damper shock
Absorbers”, Simulation Modelling Practice and Theory 17 (2009) 309–325

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