Table Of ContentSynthesis Lectures on
Mechanical Engineering
Albert C. J. Luo
Chuan Guo
Nonlinear Vibration
Reduction
An Electromagnetically Tuned
Mass Damper System
Synthesis Lectures on Mechanical
Engineering
This series publishes short books in mechanical engineering (ME), the engineering branch
that combines engineering, physics and mathematics principles with materials science to
design, analyze, manufacture, and maintain mechanical systems. It involves the production
and usage of heat and mechanical power for the design, production and operation of
machines and tools. This series publishes within all areas of ME and follows the ASME
technical division categories.
Albert C. J. Luo · Chuan Guo
Nonlinear Vibration
Reduction
An Electromagnetically Tuned Mass
Damper System
Albert C. J. Luo Chuan Guo
Department of Mechanical and Mechatronics Department of Mechanical and Mechatronics
Engineering Engineering
Southern Illinois University Southern Illinois University
Edwardsville, IL, USA Edwardsville, IL, USA
ISSN 2573-3168 ISSN 2573-3176 (electronic)
Synthesis Lectures on Mechanical Engineering
ISBN 978-3-031-17498-8 ISBN 978-3-031-17499-5 (eBook)
https://doi.org/10.1007/978-3-031-17499-5
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Preface
The vibration reduction of machines and structures is an important issue in engineering.
Once the vibration reduction of machines and structures can be achieved, the usage lives
of the machines and structures will be extended. On other hands, in recent years, one is
more interested in harvesting energy from the vibrating structures and machines. Both
vibration reduction and energy harvesting systems have the similar physical and mechan-
ical mechanisms. Thus, this book will present how to determine periodic responses in
vibration reduction systems and energy harvesting systems through a nonlinear, electro-
magnetically tuned mass damper. The tuned mass damper is one of the classic dynamic
vibration absorbers with effective devices for energy dissipation and vibration reduc-
tion. The electromagnetically tuned mass damper system is extensively used for vibration
reduction in engineering. The better understanding of the nonlinear dynamics of the elec-
tromagnetically tuned mass damper system is very important to optimize the parameters
of such systems for vibration reduction. However, until now, one cannot fully understand
complex periodic motions in such a nonlinear, electromagnetically tuned mass damper
system.
In this book, there are seven chapters. The brief literature survey will be presented
for the vibration reduction through the tuned mass damper systems. The semi-analytical
methods are presented for stable and unstable periodic motions, and the corresponding
stability and bifurcations are discussed in Chap. 2. The discretization of the tuned mass
damper systems is completed, and the methodology for period-1 and period-m motions
is presented in Chap. 3. From the discrete points, the finite Fourier series of periodic
motions is presented. In Chap. 4, the bifurcation tree of period-1 motion to chaos in the
nonlinear tuned mass damper systems is presented. The period-3 motions near the primary
period-1 motions are discussed in Chap. 5, which strongly affects on vibration reductions.
In Chap. 6, independent period-9 motions are presented, and periodic motions become
more complex. In Chap. 7, independent period-12 motions are discussed, and the periodic
motions become more and more complex. Such higher-order complex motions have their
corresponding bifurcation trees to chaos through the period-doubling bifurcations. The
semi-analytical solutions of periodic motions are presented through period-1, period-3,
period-9, and period-12 motions. The corresponding stability and bifurcations of periodic
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motions are determined. The frequency–amplitude characteristics for bifurcation routes
of such higher-order periodic motions are presented. The unstable periodic motions are
analytically determined. To keep such unstable periodic motions, some control schemes
can be embedded in such vibration reductions and energy harvesting systems.
This book helps people better understand the dynamical behaviors of an electromag-
netically tuned mass damper system for the new development and design of vibration
reduction and energy harvesting systems. The authors hope this book can provide a way
to look into nonlinear vibration reduction and energy harvesting in engineering systems.
Edwardsville, IL, USA Albert C. J. Luo
Chuan Guo
Contents
1 Introduction .......................................................... 1
2 A Semi-analytical Method .............................................. 5
3 Discretization ......................................................... 9
3.1 A Tuned Mass Damper System ..................................... 9
3.2 Period-1 Motions .................................................. 11
3.3 Period-m Motion .................................................. 15
3.4 Finite Fourier Series of Periodic Motions ............................. 17
4 Period-1 Motion to Chaos .............................................. 21
4.1 Bifurcation Routes of Period-1 Motion to Chaos ...................... 21
4.2 Frequency-Amplitude Characteristics ................................ 30
4.3 Period-1 and Period-2 Motions Illustrations ........................... 40
5 Independent Period-3 Motions .......................................... 47
5.1 Analytical Period-3 Motions ........................................ 47
5.2 Frequency-Amplitude Characteristics ................................ 51
5.3 Period-3 Motion Illustrations ....................................... 62
6 Independent Period-9 Motions .......................................... 65
6.1 A Semi-analytical Solution ......................................... 65
6.2 Frequency-Amplitude Characteristics ................................ 71
6.3 Period-9 Motion Illustrations ....................................... 78
7 Independent Period-12 Motions ......................................... 81
7.1 A Semi-analytical Solution ......................................... 81
7.2 Frequency-Amplitude Characteristics ................................ 88
7.3 Period-12 Motion Illustrations ...................................... 93
References ............................................................... 95
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About the Authors
Prof. Albert C. J. Luo has worked at Southern Illinois University Edwardsville. For
over 30 years, Dr. Luo’s contributions on nonlinear dynamical systems and mechanics
lie in: (i) the local singularity theory for discontinuous dynamical systems; (ii) dynami-
cal systems synchronization; (iii) analytical solutions of periodic and chaotic motions in
nonlinear dynamical systems; (iv) the theory for stochastic and resonant layer in nonlin-
ear Hamiltonian systems; and (v) the full nonlinear theory for a deformable body. Such
contributions have been scattered into 28 monographs and over 350 peer-reviewed journal
and conference papers. Dr. Luo served as an editor for the journal “Communications in
Nonlinear Science and Numerical Simulation”, and book series on Nonlinear Physical
Science (HEP and Springer) and Nonlinear Systems and Complexity (Springer). Dr. Luo
was an editorial member for IMeCh E Part K Journal of Multibody Dynamics and Journal
of Vibration and Control; and has also organized over 30 international symposiums and
conferences on dynamics and control.
Dr. Chuan Guo worked at Southern Illinois University Edwardsville as a research assis-
tant. Mr. Guo’s research interest lies in nonlinear vibration and discontinuous dynamics.
He has published 12 peer-reviewed journal papers, more than 5 conference articles, and
1 book chapter.
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