ADVANCES ON ANALYSIS AND CONTROL OF VIBRATIONS – THEORY AND APPLICATIONS Edited by Mauricio Zapateiro de la Hoz and Francesc Pozo
Advances on Analysis and Control of Vibrations – Theory and Applications http://dx.doi.org/10.5772/2586 Edited by Mauricio Zapateiro de la Hoz and Francesc Pozo Contributors Xingiian Jing, Wiebke Heins, Pablo Ballesteros, Xinyu Shu, Christian Bohn, Samuel da Silva, Vicente Lopes Junior, Michael J. Brennan, Grzegorz Tora, Mauricio Zapateiro, Francesc Pozo, Ningsu Luo, Hamid Reza Karimi, Tore Bakka, Hao Chen, Zhi Sun, Limin Sun, Andrés BlancoOrtega, Gerardo Silva-Navarro, Jorge Colín-Ocampo, Marco Oliver-Salazar, Gerardo VelaValdés, Bong-Jo Ryu, Yong-Sik Kong, Kongming Guo, Jun Jiang
Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book.
Publishing Process Manager Mirna Cvijic Typesetting InTech Prepress, Novi Sad Cover InTech Design Team First published August, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from
[email protected] Advances on Analysis and Control of Vibrations – Theory and Applications, Edited by Mauricio Zapateiro de la Hoz and Francesc Pozo p. cm. ISBN 978-953-51-0699-9
Contents Preface IX Section 1
New Theoretical Developments on Vibration Analysis and Control 1
Chapter 1
Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 3 Xingiian Jing
Chapter 2
LPV Gain-Scheduled Observer-Based State Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies 35 Wiebke Heins, Pablo Ballesteros, Xinyu Shu and Christian Bohn
Chapter 3
LPV Gain-Scheduled Output Feedback for Active Control of Harmonic Disturbances with Time-Varying Frequencies 65 Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn
Chapter 4
Active Vibration Control Using a Kautz Filter 87 Samuel da Silva, Vicente Lopes Junior and Michael J. Brennan
Chapter 5
The Active Suspension of a Cab in a Heavy Machine 105 Grzegorz Tora
Section 2
Vibration Control Case Studies 135
Chapter 6
On Variable Structure Control Approaches to Semiactive Control of a Quarter Car System 137 Mauricio Zapateiro, Francesc Pozo and Ningsu Luo
Chapter 7
A Computational Approach to Vibration Control of Vehicle Engine-Body Systems 157 Hamid Reza Karimi
VI
Contents
Chapter 8
Multi-Objective Control Design with Pole Placement Constraints for Wind Turbine System 179 Tore Bakka and Hamid Reza Karimi
Chapter 9
Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 195 Hao Chen, Zhi Sun and Limin Sun
Chapter 10
Automatic Balancing of Rotor-Bearing Systems 213 Andrés Blanco-Ortega, Gerardo Silva-Navarro, Jorge Colín-Ocampo, Marco Oliver-Salazar, Gerardo Vela-Valdés
Chapter 11
Dynamic Responses and Active Vibration Control of Beam Structures Under a Travelling Mass 231 Bong-Jo Ryu and Yong-Sik Kong
Chapter 12
Optimal Locations of Dampers/Actuators in Vibration Control of a Truss-Cored Sandwich Plate 253 Kongming Guo and Jun Jiang
Preface This book is a compilation of some selected articles devoted to the analysis and control of vibrations. Vibrations are a phenomenon found in many engineering systems; their harmful effects are translated into low performance, noise, energy misspend, discomfort and system breakdown, among others. These are the reasons why, in the last years, researchers have made great efforts in seeking ways to eliminate them totally or partially. The subject of vibrations has been studied for a long time. Although a wide variety of practical solutions to this problem have been found so far, several problems remain still open. Complex in nature, the ideal solution to vibration mitigation goes hand in hand with technologies that can also be mathematically and physically complex. With the advent of new technologies, sophisticated damping devices that greatly help in this work have been developed too. However, in order to make them perform optimally, new theoretical tools and deep understanding of their dynamics are required; that is why nowadays great efforts are made in this sense in this branch of the science. This book goes through some of the most recent advances in the analysis and control of vibrations. On the one hand, some chapters bring out novel theoretical developments on the analysis of vibrations; on the other hand, other chapters reveal specific applications in areas as diverse as, for instance, vehicle suspension systems, vehicle-engine-body systems, wind turbines for energy production and civil engineering structures. These pages take us up to different classical, yet effective control methodologies, as the researchers build new theories and applications upon them. The works on vibration analysis presented in this book stand out for their thoroughness in presenting the theoretical developments in order to further develop new control methodologies from these new bases. Case studies, on the other hand, stand out because they report the obtention of control systems of practical implementation; in most cases, these controllers feature an exquisite simplicity and outstanding performance ratings. These papers therefore set down a combination of rigor and simplicity which are desirable aspects in engineering.
X
Preface
This book is not intended to be a comprehensive compendium of papers on analysis and control of vibrations. However, it can become a reference for those who are getting started in this field as well as for those who already have gained experience in it. Here you will find recent developments in this regard and in addition, each chapter leads to a series of related references, making this book an important source of state of the art titles on the subject at the time this book was edited. Lastly, we the editors, want to render thanks to the authors of each chapter for the endeavor to their preparation; this book has been possible thanks to them. Mauricio Zapateiro De la Hoz and Francesc Pozo Department of Applied Mathematics III Polytechnic University of Catalonia Barcelona, Spain
Section 1
New Theoretical Developments on Vibration Analysis and Control
Chapter 1
Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain Xingiian Jing Additional information is available at the end of the chapter http://dx.doi.org/10.5772/45795
1. Introduction In the control theory of linear systems, system transfer function provides a coordinate-free and equivalent description for system dynamic characteristics, by which it is convenient to conduct analysis and design. Therefore, frequency domain methods are commonly used by engineers and widely applied in engineering practice. However, although the analysis and design of linear systems in the frequency domain have been well established, the frequency domain analysis for nonlinear systems is not straightforward. Nonlinear systems usually have very complicated output frequency characteristics and dynamic behaviour such as harmonics, inter-modulation, chaos and bifurcation. Investigation and understanding of these nonlinear phenomena in the frequency domain are far from full development. Frequency domain methods for nonlinear analysis have been investigated for many years. There are several different approaches to the analysis and design for nonlinear systems, such as describing functions [5, 13], harmonic balance [18], and frequency domain methods developed from the absolute stability theory [10], for example the well-known Popov circle theorem [12, 21] etc. Investigation of nonlinear systems in the frequency domain can also be done based on the Volterra series expansion theory [11, 15, 16, 19, 20]. There are a large class of nonlinear systems which have a convergent Volterra series expansion [2, 17]. For this class of nonlinear systems, referred to as Volterra systems, the generalized frequency response function (GFRF) was defined in [4], which is similar to the transfer function of linear systems. To obtain the GFRFs for Volterra systems described by nonlinear differential equations, the probing method can be used [16]. Once the GRFRs are obtained for a practical system, system output spectrum can then be evaluated [9]. These form a fundamental basis for the analysis of nonlinear Volterra systems in the frequency domain and provide an elegant and useful method for the frequency domain analysis of a class of nonlinear systems. Many techniques developed (e.g. the GFRFs) can be regarded as an important extension of frequency domain theories for linear systems to nonlinear cases. © 2012 Jing, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
4 Advances on Analysis and Control of Vibrations – Theory and Applications
In this study, understanding of nonlinearity in the frequency domain is investigated from a novel viewpoint for Volterra systems. The system output spectrum is shown to be an alternating series with respect to some model parameters under certain conditions. This property has great significance in that the system output spectrum can therefore be easily suppressed by tuning the corresponding parameters. This provides a novel insight into the nonlinear influence in a system. The sufficient (and necessary) conditions in which the output spectrum can be transformed into an alternating series are studied. These results are illustrated by two example studies which investigated a single degree of freedom (SDOF) springdamping system with a cubic nonlinear damping. The results established in this study demonstrate a novel characteristic of the nonlinear influence in the frequency domain, and provide a novel insight into the analysis and design of nonlinear vibration control systems. The chapter is organised as follows. Section 2 provides a detailed background of this study. The novel nonlinear characteristic and its influence are discussed in Section 3. Section 4 gives a sufficient and necessary condition under which system output spectrum can be transformed into an alternating series. A conclusion is given in Section 5. A nomenclature section which explains the main notations used in this paper is given in Appendix A.
2. Frequency response functions of nonlinear systems There are a class of nonlinear systems for which the input-output relationship can be sufficiently approximated by a Volterra series (of a maximum order N) around the zero equilibrium as [2, 17] N
y( t )
n 1
n
hn ( 1 , , n ) u(t i )d i
(1)
i 1
where hn ( 1 , , n ) is the nth-order Volterra kernel which is a real valued function of 1 , , n . For the same class of nonlinear systems, it can also be modelled by the following nonlinear differential equation (NDE) M
m
K
1 m p 0 k1 , km 0
p
c p ,m p ( k1 , , km )
K
i 1
K
d ki y(t ) dt
ki
k
m
d i u(t )
i p 1
dt ki
0
(2)
K
k where d x(t ) x(t ) , () () () , M is the maximum degree of nonlinearity dt k k 0 k1 , k p q 0 k1 0 k p q 0
in terms of y(t) and u(t), and K is the maximum order of the derivative. In this model, the parameters such as c0,1(.) and c1,0(.) are referred to as linear parameters corresponding to
d k y(t )
d ku(t )
for k=0,1,…,K; and c p ,q () for dt dt k p+q>1 are referred to as nonlinear parameters corresponding to nonlinear terms in the model p k p q ki d i y( t ) d u(t ) , e.g., y(t )p u(t )q . The value p+q is referred to as the of the form coefficients of linear terms in the model, i.e.,
i 1
dt ki
i p 1
dt ki
nonlinear degree of parameter c p ,q () .
k
and
Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 5
By using the probing method [16], a recursive algorithm for the computation of the nthorder generalized frequency response function (GFRF) for the NDE model (2) is provided in [1]. Therefore, the output spectrum of model (2) can be evaluated as [9] N
1
Y ( j )
n (2 )
n 1
n 1
n
H n ( j1 , , jn ) U ( ji )d
(3)
i 1
1 n
which is truncated at the largest order N and where, H n ( j1 , , jn )
hn ( 1 , , n )exp( j(1 1 n n ))d 1 d n
(4)
is known as the nth-order GFRF defined in [4], and hn ( 1 , , n ) is the nth-order Volterra kernel introduced in (1). When the system input is a multi-tone function described by u(t )
K
Fi cos(it Fi )
(5)
i 1
(where Fi is a complex number, Fi is the argument, Fi is the modulus, and K is a positive integer), the system output frequency response can be evaluated as [9]: N
Y ( j )
1
n n 1 2 k k 1
(6)
H n ( jk , , jk )F(k ) F(k ) n
1
1
n
n
where F ( ki ) can be explicitly written as F(k ) Fk e i
jF k sig( ki ) i
i
for k i 1, , K
in
1 a 0 stead of the form in [9], sgn( a) , and k 1 , , K . i 1 a 0 In order to explicitly reveal the relationship between model parameters and the frequency response functions above, the parametric characteristics of the GFRFs and output spectrum are studied in [6]. The nth-order GFRF can then be expressed into a more straightforward polynomial form as
Hn ( j1 , , jn ) CE Hn ( j1 , , jn ) fn ( j1 , , jn )
(7)
where CE Hn ( j1 , , jn ) is referred to as the parametric characteristic of the nth-order GFRF H n ( j1 , , jn ) , which can be recursively determined as
n 1 n q n CE( Hn ( j1 , , jn )) C0,n C p ,q CE( Hnq p 1()) C p ,0 CE( Hn p 1 ()) (8) q 1p 1 p 2 with terminating condition CE H 1 ( j i ) 1 . Note that CE is a new operator with two operations “ ” and “ ” defined in [6,7] (the definition of CE can be referred to Appendix
6 Advances on Analysis and Control of Vibrations – Theory and Applications
B and more detailed discussions in [22]), and C p , q is a vector consisting of all the (p+q)th degree nonlinear parameters, i.e., C p ,q [c p ,q (0, ,0), c p ,q (0, ,1), , c p ,q ( K , , K )] pq m
In Equation (8), fn ( j1 , , jn ) is a complex valued vector with the same dimension as CE Hn ( j1 , , jn ) . In [7], a mapping function n (CE( Hn ()); 1 , ,n ) from the parametric characteristic CE Hn ( j1 , , jn ) to its corresponding correlative function f n ( j1 , , j n ) is established as n( s ) (c p
0 , q0
()c p
1 , q1
all the 2 partitions for s satisfying s1 ( s ) c p ,q ( ) and p 0
() c p
f (c 1
k , qk
(); l(1) l( n( s )) )
p ,q (), n( s ); l(1) l( n( s )) )
all the p partitions all the different for s c p ,q ( ) permutations of {s x ,,sx } 1
p
n( s i 1
xi
( s c p ,q
f ( s s ( s c ()); xp p ,q l(1) l( n( s ) q ) ) 2 a x1
p
( ))) ( sx ( s c p ,q ());l( X ( i ) 1) l( X ( i ) n( s
xi ( s
i
cp ,q ()))) )
(9a)
where the terminating condition is k=0 and 1 (1; i ) H1( ji ) (which is the transfer function
{sx1 , sxp } is a permutation of {sx1 , sxp } ,
when all nonlinear parameters are zero),
l (1) l ( n ( s )) represents the frequency variables involved in the corresponding functions, l(i) for i=1… n(s ) is a positive integer representing the index of the frequency variables,
s cp
0 ,q0
sx ,
()c p
1 , q1
()c p
x
k , qk
() , n( sx ( s )) ( pi qi ) x 1 , x is the number of the parameters in i 1
x
( pi qi ) is the sum of the subscripts of all the parameters in s x . Moreover, i 1
i 1
X(i ) n( sx ( s c pq ()))
(9b)
j
j 1
K
Ln( j ) = c1,0 ( k1 )( j )k1
R
(9c)
k1 0
q
n( s )
k
f1 (c p ,q (), n( s ); l(1) l( n( s )) ) ( ( jl( n( s ) q i ) ) pi Ln( s ) ( j l( i ) ) f2 a ( sx sx ( s c p ,q ());l(1) l( n( s ) q ) ) 1
p
p
( jl( X(i )1) jl( X(i )n( s i 1
(9d)
i 1
i 1
xi ( s / c pq ( ))))
) ki
(9e)
The mapping function n (CE( Hn ()); 1 , ,n ) enables the complex valued function fn ( j1 , , jn ) to be analytically and directly determined in terms of the first order GFRF and nonlinear parameters. Therefore, the nth-order GFRF can directly be written into a more
Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 7
straightforward and meaningful polynomial function in terms of the first order GFRF and model parameters by using the mapping function n (CE( Hn ()); 1 , ,n ) as
Hn ( j1 , , jn ) CE Hn ( j1 , , jn ) n (CE Hn () ; 1 , ,n )
(10)
Using (10), Equation (3) can be written as Y ( j )
N
CE Hn ( j1 , , jn ) Fn ( j )
(11a)
n 1
1
where Fn ( j )
n (2 )
n 1
(6) can be written as
n
n (CE( Hn (); 1 , ,n ) U( ji )d . Similarly, Equation i 1
1 n
Y ( j )
N
CE Hn ( jk , , jk 1
n 1
where Fn ( j )
1 2 n k
1
n
(11b)
) Fn ( )
n (CE( Hn ()); k , ,k ) F(k ) F(k ) .
k
1
n
1
n
Note
that
the
n
expressions for output spectrum above are all truncated at the largest order N. The significance of the expressions in (10-11) is that, the explicit relationship between any model parameters and the frequency response functions can be demonstrated clearly and thus it is convenient to be used for system analysis and design. Example 1. Consider a simple example to demonstrate the results above. Suppose all the other nonlinear parameters in (2) are zero except c1,1(1,1), c0,2(1,1), c2,0(1,1). For convenience, c1,1(1,1) is written as c1,1 and so on. Consider the parametric characteristic of H3(.), which can easily be derived from (8), CE H 3 ( j1 , , j 3 ) 2 C 0 ,3 C 1,1 C 0 ,2 C 1,1 C 1,1 C 2 ,0 C 2 ,1 C 1,2 C 2 ,0 C 0 ,2 C 22,0 C 3,0 2 C 1,1 C 0 ,2 C 1,1 C 1,1 C 2 ,0 C 2 ,0 C 0 ,2 C 22,0
Note that C1,1= c1,1, C0,2=c0,2, C2,0=c2,0. Thus, 2 2 CE( H 3 ( j1 , , j3 )) [c1,1c0,2 , c1,1 , c1,1c2,0 , c2,0 c0,2 , c2,0 c1,1 , c2,0 ]
Using (9abc), the correlative functions of each term in CE H 3 ( j1 , , j3 ) can all be obtained. For example, for the term c1,1c0,2, it can be derived directly from (9abc) that
n( s ) (c1,1()c0,2 (); l(1) l( n( s )) ) 3 (c1,1()c0,2 (); 1 3 ) f1( c1,1(),3; 1 3 ) f2 a ( s1( c1,1()c0,2 () / c1,1 ());1 ,2 ) 2 ( s1 ( c0,2 ());1 ,2 ) f1( c1,1(),3; 1 3 ) f2 a (c0,2 ();1 ,2 ) 2 (c0,2 ();1 ,2 )
j3 j j j ( j j2 ) j1 j2 ( j1 j2 ) 1 2 3 1 L3 ( j1 j3 ) L2 ( j1 j2 ) L3 ( j1 j3 ) L2 ( j1 j2 )
8 Advances on Analysis and Control of Vibrations – Theory and Applications
Proceed with the process above, the whole correlative function of CE H3 ( j1 , , j3 ) can be obtained, and then (10-11ab) can be determined. This demonstrates a new way to analytically compute the high order GFRFs, and the final results can directly be written into a polynomial form as (10-11ab), for example in this case 2 2 H 3 ( j1 , , j3 ) [c1,1c0,2 , c1,1 , c1,1c2,0 , c2,0 c0,2 , c2,0 c1,1 , c2,0 ] 3 (CE( H 3 ( j1 , , j3 )); 1 , ,3 ) 2 2 2 2 c1,1c0,2 3 (c1,1c0,2 ; 1 , , 3 ) c1,1 3 (c1,1 ; 1 , ,3 ) ... c2,0 3 (c2,0 ; 1 , ,3 )
As discussed in [7], it can be seen from Equations (10-11ab) and Example 1 that the mapping function n (CE( Hn ()); 1 , ,n ) can facilitate the frequency domain analysis of nonlinear systems such that the relationship between the frequency response functions and model parameters, and the relationship between the frequency response functions and H1( jl(1) ) can be demonstrated explicitly, and some new properties of the GFRFs and output spectrum can be revealed. In practice, the output spectrum of a nonlinear system can be expanded as a power series with respect to a specific model parameter of interest by using (11ab) for N . The nonlinear effect on system output spectrum incurred by this model parameter which may represents the physical characteristic of a structural unit in the system can then be analysed and designed by studying this power series in the frequency domain. Note that the fundamental properties of this power series (e.g. convergence) are to a large extent dominated by the properties of its coefficients, which are explicitly determined by the mapping function n (CE( Hn ()); 1 , ,n ) . Thus studying the properties of this power series is now equivalent to studying the properties of the n (CE( Hn ()); 1 , ,n ) . Therefore, the mapping function mapping function n (CE( Hn ()); 1 , ,n ) introduced above provides an important and significant technique for this frequency domain analysis to study the nonlinear influence on system output spectrum. In this study, a novel property of the nonlinear influence on system output spectrum is revealed by using the new mapping function n (CE( Hn ()); 1 , ,n ) and frequency response functions defined in Equations (10-11). It is shown that the nonlinear terms in a system can drive the system output spectrum to be an alternating series under certain conditions when the system subjects to a sinusoidal input, and the system output spectrum is shown to have some interesting properties in engineering practice when it can be expanded into an alternating series with respect to a specific model parameter of interest. This provides a novel insight into the nonlinear effect incurred by nonlinear terms in a nonlinear system to the system output spectrum.
3. Alternating phenomenon in the output spectrum and its influence The alternating phenomena and its influence are discussed in this section to point out the significance of this novel property, and then the conditions under which system output spectrum can be expressed into an alternating series are studied in the following section.
Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 9
For any nonlinear parameter (simply denoted by c) in model (2), the output spectrum (11ab) can be expanded with respect to this parameter into a power series as Y ( j ) F0 ( j ) cF1 ( j ) c 2 F2 ( j ) c F ( j )
(12)
Note that when c represents a nonlinearity from input terms, Equation (12) may be a finite series; in other cases, it is definitely an infinite series, and if only the first terms in the series (12) are considered, there is a truncation error denoted by ( ) . As demonstrated in Example 1, Fi ( j ) for i=0,1,2,… are some scalar frequency functions and can be obtained ~ from Fi ( j ) or Fi ( j ) in (11a,b) by using the mapping function n (CE( Hn ()); 1 , ,n ) . Clearly, Fi ( j ) dominates the fundamental properties of this power series such as convergence. Thus these properties of this power series can be revealed by studying the property of n (CE( Hn ()); 1 , ,n ) . This will be discussed more in the next section. In this section, the alternating phenomenon of this power series and its influence are discussed.
For any ℂ, define an operator as
sgnc ( ) sgn r (Re( )) sgnr (Im( )) 1 x0 where sgn r ( x) 0 x 0 for x ℝ. 1 x0
Definition 1 (Alternating series). Consider a power series of form (12) with c>0. If sgn c ( Fi ( j )) sgn c ( Fi 1( j )) for i=0,1,2,3,…, then the series is an alternating series.
The series (12) can be written into two series as
Y ( j ) Re(Y ( j )) j(Im(Y( j ))) Re( F0 ( j )) c Re( F1( j )) c 2 Re( F2 ( j )) c Re( F ( j )) 2
(13)
j(Im( F0 ( j )) c Im( F1 ( j )) c Im( F2 ( j )) c Im( F ( j )) ) From definition 1, if Y ( j ) is an alternating series, then Re(Y ( j )) and Im(Y ( j )) are both alternating. When (12) is an alternating series, there are some interesting properties summarized in Theorem 1. Denote Y ( j )1 F0 ( j ) cF1 ( j ) c 2 F2 ( j ) c F ( j )
Theorem 1. Suppose (12) is an alternating series at a ( ℝ+) for c>0, then:
(1) if there exist T>0 and R>0 such that for i>T Im( Fi ( j )) Re( Fi ( j )) min , R Re( F ( j )) Im( Fi 1 ( j )) i 1
(14)
10 Advances on Analysis and Control of Vibrations – Theory and Applications
then (12) has a radius of convergence R, the truncation error for a finite order ( ) c 1 F 1 ( j ) , and for all n 0,
>T is
Y ( j ) n [ Y ( j )1T 2 n 1 , Y ( j )1T 2 n ] and n 1 n ; 2
( j ) Y( j )Y( j ) is also an alternating series with respect to parameter c; (2) Y 2 ( j ) Y( j )Y( j ) is alternating only if Re(Y ( j )) is alternating; Furthermore, Y (3) there exists a constant c 0 such that
Y ( j ) c
0 for 0 c c .
Proof. See Appendix C.□ The first point in Theorem 1 shows that only if there exists a positive constant R>0, the series must be convergent under 0
