Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay
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x Contents
29 Complex Frequency Analysis of an Axially Moving String
with Multiple Attached Oscillators by Using Green’s
Function Method .............................................................371
Le-Feng L¨u, Yue-Fang Wang, and Ying-Xi Liu
30 Model Reduction on Inertial Manifolds of Navier–Stokes
Equations Through Multi-scale Finite Element ...........................383
Jia-Zhong Zhang, Sheng Ren, and Guan-Hua Mei
31 Diesel Engine Condition Classification Based
on Mechanical Dynamics and Time-Frequency Image
Processing .....................................................................397
Hongkun Li and Zhixin Zhang
32 Input Design for Systems Under Identification as Applied
to Ultrasonic Transducers ...................................................409
Nishant Unnikrishnan, Yicheng Pan, Marco Schoen,
Ajay Mahajan, Jarlen Don, and Tsuchin Chu
33 Development of a Control System for Automating of Spiral
Concentrators in Coal Preparation Plants.................................421
Josh Hoelscher, Yicheng Pan, Manoj Mohanty, Jarlen Don,
Tsuchin Chu, and Ajay Mahajan
34 On the Rough Number Computation and the Ada
Language ......................................................................437
Trong Wu
Author Index........................................................................449
Subject Index .......................................................................461
Contributors
Deng-Qing Cao School of Astronautics, Harbin Institute of Technology,
P.O. Box 137, Harbin 150001, People’s Republic of China, dqcao@hit.edu.cn
Tsuchin Chu Department of Mechanical Engineering and Energy Processes,
Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale,
IL 62901-6603, USA, tchu@siu.edu
Jarlen Don Department of Mechanical Engineering and Energy Processes,
Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale,
IL 62901-6603, USA
Dejun Fan MOE Key Laboratory for Strength and Vibration, School of Aerospace,
Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China
and
Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai,
Shandong 264209, People’s Republic of China, fandejun@hitwh.edu.cn
S.-B. Fan CSR Qingdao Sifang Locomotive and Rolling Stock Company,
Chengyang, Qingdao 266111, People’s Republic of China
Ruicheng Feng School of Mechano-Electronic Engineering, Lanzhou University
of Technology, Lanzhou 730050, People’s Republic of China
Xilin Fu School of Mathematical Sciences, Shandong Normal University,
Jinan 250014, People’s Republic of China, xilinfu@gmail.com
Brandon C. Gegg Mechanical Engineering, Texas A&M University, College
Station, TX 77843, USA, barracuda@tamu.edu
Yu Guo Department of Mechanical and Industrial Engineering, Southern Illinois
University Edwardsville, Edwardsville, IL 62026-1805, USA, yguo@siue.edu
H.R. Hamidzadeh Department of Mechanical and Manufacturing Engineering,
Tennessee State University, Nashville, TN, USA, HHamidzadeh@tnsate.edu
Xiuping Han School of Mathematical Sciences, Shandong Normal University,
Jinan 250014, People’s Republic of China, han027@163.com
xi
xii Contributors
Josh Hoelscher Department of Mechanical Engineering and Energy Processes,
Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL
62901-6603, USA
Ling Hong MOE Key Lab for Strength and Vibration, School of Aerospace, Xi’an
Jiaotong University, Xi’an 710049, People’s Republic of China, hongling@mail.
xjtu.edu.cn
Wen-Hu Huang School of Astronautics, Harbin Institute of Technology,
P.O. Box 137, Harbin 150001, People’s Republic of China
Yaping Huang Department of Coatings and Polymeric Materials, North Dakota
State University, Fargo, ND 58108, USA, yaping.huang@ndsu.edu
Jun Jiang MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong
University, Xi’an 710049, People’s Republic of China, jun.jiang@mail.xjtu.edu.cn
Wuyin Jin School of Mechano-Electronic Engineering, Lanzhou University
of Technology, Lanzhou 730050, People’s Republic of China
and
Key Laboratory of Digital Manufacturing Technology and Application,
The Ministry of Education, Lanzhou University of Technology, Lanzhou 730050,
People’s Republic of China, jinwuyin@263.net
Yanfei Jin Department of Mechanics, Beijing Institute of Technology, Beijing
100081, People’s Republic of China, jinyf@bit.edu.cn
K. Kappaganthu Department of Mechanical Engineering, Villanova University,
Villanova, PA, USA, karthik.kappaganthu@villanova.edu
Timo Karvinen Department of Mechanics and Design, Tampere University
of Technology, P.O. Box 589, 33101 Tampere, Finland, timo.karvinen@tut.fi
Erno Keskinen Department of Mechanical Engineering, Tampere University
of Technology, P.O. Box 589, 33101 Tampere, Finland, erno.keskinen@tut.fi
Peng-Fei Lei School of Energy and Power Engineering, Xi’an Jiaotong University,
Xi’an, Shaanxi 710049, People’s Republic of China, pflei@stu.xjtu.edu.cn
Yonggang Leng School of Mechanical Engineering, Tianjin University, Tianjin
300072, People’s Republic of China, Hleng
yg@tju.edu.cn
Hongkun Li School of Mechanical Engineering, Dalian University of Technology,
Dalian 116023, People’s Republic of China
and
State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian
116023, People’s Republic of China, lihk@dlut.edu.cn
Kai-Lun Li School of Energy and Power Engineering, Xi’an Jiaotong University,
Xi’an, Shanxi 710049, People’s Republic of China, likailun@stu.xjtu.edu.cn
Hao Liu School of Mechano-Electronic Engineering, Lanzhou University
of Technology, Lanzhou 730050, People’s Republic of China
Contributors xiii
Yan Liu School of Mechatronics, Northwestern Polytechnical University, Xi’an
710072, People’s Republic of China, liuyan@nwpu.edu.cn
Ying-Xi Liu Department of Engineering Mechanics, Dalian University
of Technology, 2 Linggong Road, Dalian 116024, People’s Republic of China,
yxliu@dlut.edu.cn
Jing-jun Lou College of Naval Architecture and Power, Naval University
of Engineering, Wuhan 430033, People’s Republic of China,
jingjun
lou@hotmail.com
Le-Feng L¨u Department of Engineering Mechanics, Dalian University
of Technology, 2 Linggong Road, Dalian 116024, People’s Republic of China,
lorrykey@gmail.com
Albert C.J. Luo Department of Mechanical and Industrial Engineering,
Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA,
aluo@siue.edu
Yajun Luo School of Aerospace, Xi’an Jiaotong University, Xi’an 710049,
People’s Republic of China, luoyajun@mail.xjtu.edu.cn
Ajay Mahajan Department of Mechanical Engineering, University of Akron,
Akron, OH, USA
Guan-Hua Mei School of Energy and Power Engineering, Xi’an Jiaotong
University, No. 28, Xianning West Road, Xi’an, Shaanxi 710049, People’s
Republic of China, meiguanhua@stu.xjtu.edu.cn
Manoj Mohanty Department of Mining and Mineral Resources Engineering,
Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL
62901-6603, USA
Jori Montonen Department of Mechanics and Design, Tampere University of
Technology, P.O. Box 589, FIN-33101 Tampere, Finland
C. Nataraj Department of Mechanical Engineering, Villanova University,
Villanova, PA, USA, nataraj@villanova.edu
Yicheng Pan Department of Mechanical Engineering and Energy Processes,
Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale,
IL 62901-6603, USA
J.H. Peng Department of Mechanical Engineering, Shaoyang Polytechnic, Hunan,
People’s Republic of China
and
Department of Physics, Shaoyang University, Shaoyang, Hunan, People’s Republic
of China, pengjhua@sina.com
Neil Popplewell Mechanical and Manufacturing Engineering, University
of Manitoba, 15 Gillson Street, Winnipeg, MB, Canada R3T 5V6,
npopple@cc.umanitoba.ca
xiv Contributors
Rajkumar Porwal Department of Mechanical Engineering, Shri G. S. Institute of
Technology and Science, 23 Park Road, Indore 452003, India, rkporwal@iitk.ac.in,
rajkumarporwal@hotmail.com
Sheng Ren School of Energy and Power Engineering, Xi’an Jiaotong University,
No. 28, Xianning West Road, Xi’an, Shaanxi 710049, People’s Republic of China,
sheng.ren@stu.xjtu.edu.cn
Marco Schoen Department of Mechanical Engineering, Idaho State University,
Pocatello, ID, USA
Arvind H. Shah Civil Engineering, University of Manitoba, 15 Gillson Street,
Winnipeg, MB, Canada R3T 5V6, shah@cc.umanitoba.ca
Huilin Shang School of Mechanical and Automation Engineering, Shanghai
Institute of Technology, Shanghai 20035, People’s Republic of China,
suliner60@hotmail.com
Zhiyong Shang MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong
University, Xi’an, People’s Republic of China, zhiyongtop@163.com
Bo Song School of Engineering, University of California, Merced CA 95344,
USA
Darryl K. Stoyko Mechanical and Manufacturing Engineering, University
of Manitoba, 15 Gillson Street, Winnipeg, MB, Canada R3T 5V6,
D
Stoyko@UManitoba.ca
Jian-Qiao Sun School of Engineering, University of California, Merced,
CA 95344, USA, jqsun@ucmerced.edu
Steve C.S. Suh Mechanical Engineering, Texas A&M University, College Station,
TX 77843, USA
J.S. Tang College of Mechanics and Aerospace, Hunan University, Changsha,
Hunan, People’s Republic of China, js.tang@yahoo.com.cn
X.W. Tangpong Department of Mechanical Engineering, North Dakota State
University, Fargo, ND 58108, USA, annie.tangpong@ndsu.edu
Nishant Unnikrishnan Department of Mechanical Engineering and Energy
Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive,
Carbondale, IL 62901-6603, USA
Nalinaksh S. Vyas Department of Mechanical Engineering, Indian Institute
of Technology Kanpur, Kanpur 208016, India, vyas@iitk.ac.in
Jin-Lin Wang School of Astronautics, Harbin Institute of Technology,
P.O. Box 137, Harbin 150001, China
X. Wang Institute of Science, PLA University of Science and Technology,
211101 Nanjing, People’s Republic of China
Contributors xv
Yang Wang Southern Illinois University Edwardsville, Edwardsville,
IL 62026-1805, USA, ywang@siue.edu
Ying-chun Wang Administrative Office of Training, Naval University
of Engineering, Wuhan 430033, People’s Republic of China
Yue-Fang Wang Department of Engineering Mechanics, Dalian University
of Technology, 2 Linggong Road, Dalian 116024, People’s Republic of China
and
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian
116024, People’s Republic of China, yfwang@dlut.edu.cn
Z.H. Wang Institute of Science, PLA University of Science and Technology,
211101 Nanjing, People’s Republic of China
and
Institute of Vibration Engineering Research, Nanjing University of Aeronautics and
Astronautics, Nanjing 210016, People’s Republic of China, zhwang@nuaa.edu.cn
Trong Wu Department of Computer Science, Southern Illinois University
Edwardsville, Edwardsville, IL 62026-1656, USA, twu@siue.edu
Ying Wu School of Science, Xi’an University of Technology, Xi’an, Shaanxi
710054, People’s Republic of China, wying36@163.com
Minglong Xu School of Aerospace, Xi’an Jiaotong University, Xi’an 710049,
People’s Republic of China, mlxu@mail.xjtu.edu.cn
Y. Xu CSR Qingdao Sifang Locomotive and Rolling Stock Company, Chengyang,
Qingdao 266111, People’s Republic of China
Aihua Zhang College of Electrical and Information Engineering, Lanzhou
University of Technology, Lanzhou 730050, People’s Republic of China
Chi Zhang School of Mechano-Electronic Engineering, Lanzhou University of
Technology, Lanzhou 730050, People’s Republic of China
H. Zhang CSR Qingdao Sifang Locomotive and Rolling Stock Company,
Chengyang, Qingdao 266111, People’s Republic of China
Jia-Zhong Zhang School of Energy and Power Engineering, Xi’an Jiaotong
University, Xi’an, Shaanxi 710049, People’s Republic of China,
jzzhang@mail.xjtu.edu.cn
Xinong Zhang School of Aerospace, Xi’an Jiaotong University, Xi’an 710049,
People’s Republic of China, xnzhang@mail.xjtu.edu.cn
Y.-W. Zhang MOE Key Lab for Strength and Vibration, Xi’an Jiaotong University,
Xi’an 710049, People’s Republic of China, bee.bean4@stu.xjtu.edu.cn
Zhixin Zhang School of Mechanical Engineering, Dalian University
of Technology, Dalian 116023, People’s Republic of China, dlzzx@dl.cn
xvi Contributors
Y.-J. Zhao CSR Qingdao Sifang Locomotive and Rolling Stock Company,
Chengyang, Qingdao 266111, People’s Republic of China,
zhaoyanju215@yahoo.com.cn
P. Zhu Department of Physics, Simao Teacher’s College, Puer 665000, People’s
Republic of China, zhuupp@yahoo.com.cn
Shi-jian Zhu College of Naval Architecture and Power, Naval University
of Engineering, Wuhan 430033, People’s Republic of China
Part I
Nonlinear and Discontinuous Dynamical
Systems
Chapter 1
General Solution of a Vibration System with
Damping Force of Fractional-Order Derivative
Z.H. Wang and X. Wang
Abstract The generalized Bagley–Torvik system is a linear oscillator whose
damping force is described by fractional-order derivative with order between
0 and 2. This paper shows that as a sequential fractional-order differential equation
with constant coefficients whose general solution depends on more than two free
(independent) constants, the generalized Bagley–Torvik equation actually admits a
general solution that involves two free constants only and can be determined fully
by the initial displacement and initial velocity.
1.1 Introduction
Dynamical systems with fractional-order derivatives have found many applications
in various problems in science and engineering such as viscoelasticity, heat con-
duction, electrode–electrolyte polarization, electromagnetic waves, diffusion wave,
control theory and so on [1–5]. In order to model the forced motion of a rigid plate
immersed in a kind of Newtonian fluid, for example, Bagley and Torvik proposed
the following differential equation [5]
A Rx.t/ C B
0
D
3=2
x.t/ C Cx.t/ D f.t/: (1.1)
A peculiarity of this equation is the fractional-order derivative B
0
D
3=2
x.t/ that is
used to describe the damping force. Recently, the first author proved in [6] that the
fractional-order derivative B
0
D
˛
x.t/ appeared in
Z.H. Wang (
)
Institute of Science, PLA University of Science and Technology, 211101 Nanjing,
People’s Republic of China
and
Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics,
210016 Nanjing, People’s Republic of China
e-mail: zhwang@nuaa.edu.cn
A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,
DOI 10.1007/978-1-4419-5754-2
1,
c
Springer Science+Business Media, LLC 2010
3
4 Z.H. Wang and X. Wang
A Rx.t/ C B
0
D
˛
x.t/ C Cx.t/ D 0.A;B;C>0I 0<˛<2/ (1.2)
acts always as a damping force, so that the motion governed by (1.2)isaresultof
external force, elastic restoring force and fractional-order damping force. A funda-
mental problem is how to formulate the initial conditions so that the evolution of
the system is completely determined from the initial conditions [8]. Because the or-
der of highest derivative in (1.2) is 2, it is expected that as in the case of classical
vibration systems, the initial position and initial velocity determine completely the
evolution of the vibration systems.
A straightforward approach for solving this linear equation (1.2) is Laplacian
transformation [1,7]. In this way, a solution in terms of the initial position and initial
velocity can be easily obtained by using this method, if the Caputo’s fractional-order
derivative is employed. However, in [8], the solution of the vibration system has to
use the value of a fractional-order derivative at the initial point, except for the initial
position and initial velocity, if the Riemann–Liouville’s fractional-order derivative
is applied. Recently, Bonilla et al. established a theory for finding the general so-
lution of sequential fractional-order differential equations (SFDEs) with constant
coefficients [9]. As in solving linear ordinary differential equations with constant co-
efficients, the general solution of a SFDE can be obtained directly by the roots of
a polynomial (be called characteristic roots for simplicity). This paper studies the
general solution of the generalized Bagley–Torvik equation in dimensionless form
Rx.t/ C
0
D
˛
x.t/ C x.t/ D 0.>0;0<˛<2;˛¤ 1/: (1.3)
Regarding the fractional vibration equation as a SFDE, its “order” is greater than 2,
and it requires more than two free constants to describe its general solution. The
main objective of this paper is to show that with Riemann–Liouville’s derivative or
Caputo’s derivative, the general solution of (1.3) involves two free constants only,
and it can be determined fully by the initial displacement and initial velocity.
1.2 Preliminaries for Sequential Fractional-Order
Differential Equations
The following homogeneous linear fractional differential equation
"
a
D
n˛
C
n1
X
kD0
a
k
.t/
a
D
k˛
#
x.t/ D 0 (1.4)
is called SFDE, where a; a
k
2 R,and
a
D
k˛
D
a
D
˛
.
a
D
.k1/˛
/.kD 2;3;:::;n/:
The fractional-order derivative
a
D
˛
can be defined in different ways [1,3], including
Riemann–Liouville’s definition.