vibration

1. Dynamical analysis of nonlinear quarter car models considering
fractional order damping with sprung and unsprung mass
By
Tadios Molla
A Synopsis report submitted in partial fulfilment of the requirements for
the degree of Doctor of Philosophy in Production Engineering
Under Supervision of
Dr. Karthikeyan R.,
Professor, Center for Nonlinear systems, Chennai Institute of Technology
(CIT).
and Co-Supervision of
Dr. Prakash D.,
Center for Nonlinear systems, Chennai Institute of Technology (CIT).
November 2023.

2. Presentation outline
 Introduction
 Literature review
 Problem statement
 Research Objective
 Methodology
Work description
 Contributions and findings
 Conclusion and recommendation
 publication
 References
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3. • The need for passenger comfort, safety, and stability in cars is growing.
• Study of Suspension system aims at benefiting-
• Ride comfort
• Stability of vehicle
fig.1 vehicle suspension
1. INTRODUCTION
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• The quarter car model is a simplified representation of a vehicle's
suspension system
• The model consists of the following components: the sprung mass, and
unsprung mass.
fig.2 vehicle suspension

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1.1 Nonlinear Elements in the Quarter Car Model
• In order to accurately represent actual vehicle suspensions, it is essential
to consider non-linear components.
• The following elements introduce complexities
a) Spring stiffness
b) Damper
c) Irregular Response
• Understanding and accurately modeling these nonlinear elements is
crucial for ensuring both ride comfort and safety.
• In our analysis, we've considered the nonlinear elements in the quarter
car model to create a more realistic representation of how vehicles
behave on real-world roads.

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1.2 Fractional Order (FO) application in the Model
• ODEs (e.g., first-order or second-order), may not be able to specify
the complicated dynamical behaviors and many hidden physical
features of nonlinear chaotic oscillators.
• FO systems exhibit memory effects, meaning they consider past
states when determining future behavior.
• Using FO, we can capture the system's behavior more effectively,
especially when dealing with real-world road conditions and
disturbances.
• FO is an effective instrument to model actual scenarios with
complicated physical dynamics accurately.

7.  There are several definitions of the Fractional derivative:
 Riemann-Liouville derivative,
 Grünwald-Letnikov derivative,
 Caputo derivative
• The initial conditions for the fractional-order differential
equations in the Caputo sense are the same as like their
integer-order counter parts.,
• Hence It is Suitable to apply initial conditions with physical
systems.
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8. 2. Literature Review
One degree of freedom
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9. Two degree of freedom
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10. Summary of Literature Review
The review of related literature can be summarized and as follows:
• Numerous studies rely on assuming linear spring characteristics.
 Most of the research solutions are obtained using integer order.
 The majority of research solutions are found using numerical
methods.
 Periodic and quasi-periodic road profiles are emphasized by
several researchers as a means of stimulation.

11. 3. Statement of the Problem
• These days, accurate modeling of these systems has become essential.
• The IO is not able to clearly depict the dynamic behavior or disclose the
true physical properties.
• The suspension system's non-linear features are not adequately handled.
• Lack of algorithms to examine the incommensurate fractional order
system's dynamical characteristics.
• It is necessary to reevaluate the chaotic response of the current quarter
vehicle models in light of a considerably more precise numerical analysis
made possible by fractional calculus.
This study aims to address this gap and evaluating their potential for
improving ride quality, and safety. 11
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12. 4. Objectives
4.1General Objective
• The main objective of this research work is to analyze Dynamical
analysis of nonlinear quarter car models considering fractional
order damping with sprung and unsprung masses exposed to
various road profiles.
4.2Specific Objective
• To analyze integer order (IO) nonlinear quarter car model.
• To derive and analyze (n-DOF) nonlinear model of a quarter car
with fractional order
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13. • To investigate the response for FO system under different
excitations (Periodic, Quasi-Periodic & Stochastic).
• To develop numerical algorithms and methods to analyze
the stability of an incommensurate FO nonlinear system.
• To conduct analytical and numerical analysis n-DOF FO
nonlinear quarter car models described by nonlinear spring
and damping
• To conduct experimental validation
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5. Research Methodology
fig.3 methodology flow chart

15. 6. Work description
6.1 Dynamic Analysis
6.1.1 mathematical modelling
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fig.4 one degree of freedom
quarter car models
• IO system's state space representation
• system dynamical equation [39]
Where: Fc is hysteretic nonlinear damping and
stiffness force
Where: F(t) is road excitation
• FO system's state space representation

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Fig.5 2DoF
• system dynamical equation [35]
Where: xo is road excitation
• FO system's state space representation
• IO system's state space representation

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6.1.2 Stability analysis
• satisfy the stability condition for chaotic behavior.
one degree of freedom quarter car models
Two degrees of freedom quarter car models

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 The LEs of SDoF for fractional order α =0.998 is calculated as
L1 = −0.5411, L2 = −0.5413, L3 = 0.7675,
L4 = −0.0594 and L5 = −7.1908.
• Since the system has one positive Lyapunov exponent, it is categorized
as chaotic system and L1 + L2 + L3 + L4 + L5 = −8.3326 < 0, this
shows that the system is dissipative.
 The LEs of 2DoF for fractional order α =0.998 is calculated as
L1 = 0, L2 = −1.886, L3 = -1.867, L4 = −3.232,
L5 = 1.066, L6 = −0.258 and L7 = −7.162
• Since the system has one positive Lyapunov exponent, it is categorized
as chaotic system and L1+L2+L3+L4 +L5 + L6 + L7 = −13.319 < 0,
this shows that the system is dissipative.

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Fig. 6 Time series plot and Phase Portrait for IO
SDoF for FO
SDoF for IO
 The system exhibits
chaotic behavior.
Fig. 7 FO for different values of α=0.88,0.90,0.95,0.998
6.1.3 Phase portrait

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• For α=0.88, a clear periodic oscillation and while increasing the α value
model exhibits chaotic oscillations.
• These findings confirm that fractional-order models provide a more
accurate representation of nonlinear systems with complex dynamics
compared to their integer-order counterparts.
Fig. 8 values of α=0.88,0.998

21. • Put here the commensurate and
incommensurate phases
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22. • Do the same for 2dof below
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Fig. 11 Time series plot and Phase Portrait for the system (5)
of SDoF for FO
of SDoF for IO
the system exhibits chaotic behaviour.
Fig. 12 Dynamical behavior of the state variables of the syste
=0.88,0.90,0.92,0.95,0.98,0.99

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• We can observe that for α=0.88, a clear periodic oscillation and
while increasing the α value model exhibits chaotic oscillations.
These findings confirm that fractional-order models provide a
more accurate representation of nonlinear systems with complex
dynamics compared to their integer-order counterparts.

25. • 2DOF Put here the commensurate and
incommensurate phases
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6.1.4 Bifurcation Plot & Lyapunov Spectrum
• Investigate bifurcation diagrams by varying the parameters q and b.
• In Figure 4, it is evident that within the range of 0.895 ≤ α ≤ 0.9169, the
system exhibits period doubling,
• while for α values in the range of 0.917 ≤ α ≤ 1.000, the suspension
system displays chaotic vibrations.

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• The fractional order of the system is fixed at α = 0.998.
• By fixing all the other parameters, and “b” varied.
The system indicates a period doubling bifurcation at b = 4.653 and
two regions of chaotic oscillations for 4.83 ≤ b ≤ 5.3, 5.549 ≤ b ≤ 7.
Fig. (a)bifurcation and (b) Lyapunov spectrum

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6.2 Experimental Validation
• Conducted physical experiments on quarter car setups equipped with
sensors and collect empirical data.
Fig 1. Experimental setup

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• The following experimental setup is used to conduct the testing, it
consists of (i) Vibration Exciter, (ii) Springs, (iii) Magnetorhelogical
damper, (iv) Vibration sensors (vi) Data Acquisition system (vii)
Software for visualization.
Quarter car model for experimental validation

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Fig 3. Response of the system
• The time domain response provides insights into the system's transient
behavior and damping effects over time.
• The frequency domain response analyzes the system's behavior in terms
of frequency components and resonances.

31. • Relation between experiment and IO simulation and FO
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• Relation between experiment and FO simulation

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7. contributions and findings
Here are the major contributions and findings in this research study:
• Developed a fractional order nonlinear quarter car suspension system
using the Caputo derivative.
• Stability analyses of integer and fractional orders were performed.
• derived the Adam-Bashforth-Moulton (ABM) numerical approach for
the fractional order quarter car model.
• Completed numerical simulations for both integer and fractional orders
at one and two degrees of freedom using various road excitations (periodic,
quasi-periodic, and stochastic).
• A comparison was also made between the numerical simulations and
experimental validation.

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8. Outline of the thesis structure
The structure of this research project can be summarized as follows: In
chapter 1, we provide an introduction to the research, discussing its
background, objectives, significance, and the scope and limitations of the
study. Chapter 2 delves into a review of relevant literature concerning
quarter car models with one and two degrees of freedom. Chapter 3
outlines the theoretical framework and methodology used in this study,
while Chapter 4 presents the results of numerical simulations. In Chapter 5,
we examine the experimental validation results, followed by a discussion of
these results in Chapter 6. Finally, in Chapter 7, we summarize the key
conclusions, contributions, and propose potential avenues for future
research

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9. Results and conclusion
• FO model uncovered the intricate dynamic characteristics remained
obscured when ordinary differential equations were employed.
• We explore the diverse dynamic attributes linked with integer and
fractional order systems through the examination of equilibrium
stability, Lyapunov exponents, phase portraits, bifurcations, and
Lyapunov spectra.
• The analysis emphasizing the superiority of utilizing fractional order to
depict these dynamic attributes.

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• Fractional order makes clear chaotic region at an earlier stage compared
to integer order.
• This research established a significant correlation between experimental
validation and the application of fractional order analysis.
• Only an overview of some of the findings and conclusions is provided
in this synopsis.
• The thesis presents comprehensive findings and analysis along with
values, figures, plots, and conclusions.

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10. publications
1.

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[5] D. J. Li, “Adaptive output feedback control of uncertain nonlinear chaotic systems based on
dynamic surface control technique,” Nonlinear Dynamics, vol.68, no.1-2, 10.1007/s11071-011-
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40. Thank you!!
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