**Meet the editors**

Dr Mauricio Zapateiro received his BSc in Electronics Engineering in 2005 from Universidad del Norte (Barranquilla, Colombia), and his PhD in Information Technology in 2009 from the University of Girona (Girona, Spain). In 2010, he joined the Department of Applied Mathematics III and the College of Industrial Engineering at Universitat Politècnica de Catalunya-Bar-

celonaTECH (Barcelona, Spain) where he currently is a "Juan de la Cierva" researcher. His main research interests are the control of complex systems involving nonlinearities, parametric uncertainties and unknown disturbances with applications to the active and semiactive control of vibrations in civil engineering structures, marine structures and mechanical systems such as vehicle suspensions. The contributions of his work have been published in over 25 journal papers and several conference talks.

Dr Francesc Pozo received the degree in Mathematics from the University of Barcelona (UB), Spain, in 2000 and PhD in Applied Mathematics in 2005 from the Technical University of Catalonia (UPC), Spain. He is currently with the Department of Applied Mathematics III and the College of Industrial Engineering of Barcelona (EUETIB) (since 2000) at Universitat Politècnica de Cata-

lunya-BarcelonaTECH, Barcelona, Spain, where he belongs to the research group Control, Dynamics and Applications (www-ma3.upc.edu/codalab). He is also a teaching collaborator at the Open University of Catalonia (UOC). His research interests include modelling, dynamics, control theory and applications, particularly in civil engineering, like vibration control of structures. Dr Pozo, currently an Associate Professor, has published over 20 papers on these topics.

Contents

**Preface IX** 

Xingiian Jing

Grzegorz Tora

**Section 2 Vibration Control Case Studies 135** 

Chapter 6 **On Variable Structure Control Approaches** 

Chapter 7 **A Computational Approach to Vibration** 

Hamid Reza Karimi

**Section 1 New Theoretical Developments** 

Chapter 1 **Vibration Control by Exploiting** 

**on Vibration Analysis and Control 1** 

**Feedback for Active Control of Harmonic** 

Chapter 4 **Active Vibration Control Using a Kautz Filter 87** 

Chapter 2 **LPV Gain-Scheduled Observer-Based State** 

**Nonlinear Influence in the Frequency Domain 3** 

**Disturbances with Time-Varying Frequencies 35** 

Chapter 3 **LPV Gain-Scheduled Output Feedback for Active Control of** 

Chapter 5 **The Active Suspension of a Cab in a Heavy Machine 105** 

Wiebke Heins, Pablo Ballesteros, Xinyu Shu and Christian Bohn

**Harmonic Disturbances with Time-Varying Frequencies 65**  Pablo Ballesteros, Xinyu Shu, Wiebke Heins and Christian Bohn

Samuel da Silva, Vicente Lopes Junior and Michael J. Brennan

**to Semiactive Control of a Quarter Car System 137**  Mauricio Zapateiro, Francesc Pozo and Ningsu Luo

**Control of Vehicle Engine-Body Systems 157** 

## Contents

### **Preface XI**


Hamid Reza Karimi


## 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

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

Lastly, we the editors, want to render thanks to the authors of each chapter for the

**Mauricio Zapateiro De la Hoz and Francesc Pozo** 

Department of Applied Mathematics III Polytechnic University of Catalonia

> Barcelona, Spain

endeavor to their preparation; this book has been possible thanks to them.

the art titles on the subject at the time this book was edited.

**Section 1** 

**New Theoretical Developments** 

**on Vibration Analysis and Control** 

**New Theoretical Developments on Vibration Analysis and Control** 

**Chapter 1** 

© 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.

© 2012 Jing, licensee InTech. This is a paper 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.

extension of frequency domain theories for linear systems to nonlinear cases.

**Vibration Control by Exploiting Nonlinear** 

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

**Influence in the Frequency Domain** 

Additional information is available at the end of the chapter

Xingiian Jing

http://dx.doi.org/10.5772/45795

**1. Introduction** 

## **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. © 2012 Jing, licensee InTech. This is a paper 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.

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]

$$y(t) = \sum\_{n=1}^{N} \int\_{-\infty}^{\infty} \cdots \int\_{-\infty}^{\infty} h\_n(\tau\_1, \cdots, \tau\_n) \prod\_{i=1}^{n} u(t - \tau\_i) d\tau\_i \tag{1}$$

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 5

(3)

   

is the *n*th-order Volterra

 

 

(5)

 

(6)

for *ki* ,,1 *K* in

By using the probing method [16], a recursive algorithm for the computation of the *n*thorder generalized frequency response function (GFRF) for the NDE model (2) is provided in

> 1 1 1 1 <sup>1</sup> ( ) ( ,, ) ( )

<sup>1</sup> <sup>1</sup> 1 1 <sup>1</sup> ( ,, ) ( , , )exp( ( )) *Hj j <sup>n</sup> <sup>n</sup> n n n n <sup>n</sup>*

 

( ) cos( )

(where *Fi* is a complex number, *<sup>i</sup> F* is the argument, *Fi* is the modulus, and *K* is a positive

<sup>1</sup> ( ) ( , , )( ) ( ) <sup>2</sup> *n n*

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 *n*th-order GFRF can then be expressed into a more straightforward

1 11 ( ,, ) ( ,, ) ( ,, ) *H j j CE H j j f j j n n n nn n*

<sup>1</sup> 0, , <sup>1</sup> ,0 <sup>1</sup> 1 1 <sup>2</sup> ( ( , , )) ( ( )) ( ( )) *n n n q*

*C CE H*

*<sup>n</sup> n n pq n q p <sup>p</sup> n p q p <sup>p</sup>*

operations " " and " " defined in [6,7] (the definition of *CE* can be referred to Appendix

 

, which can be recursively determined as

*Y j Hj j F F*

*ut F t F* 

*i ii*

*n nk k k k*

*<sup>k</sup> <sup>k</sup> F Fe* 

, and <sup>1</sup> , , *<sup>i</sup>*

*<sup>k</sup>* 

is referred to as the parametric characteristic of the *n*th-order

(8)

 

*<sup>i</sup>* . Note that *CE* is a new operator with two

 

*n n ni*

 

 

1 1

( ) *<sup>k</sup> <sup>i</sup> <sup>i</sup> i i*

*j F k*

 *<sup>K</sup>* .

 (7)

 

 *<sup>j</sup> d d* (4)

*N n*

*n i Y j H j j Uj d*

[1]. Therefore, the output spectrum of model (2) can be evaluated as [9]

*n*

*h*

which is truncated at the largest order *N* and where,

is known as the *n*th-order GFRF defined in [4], and ),,( *<sup>n</sup>* <sup>1</sup> *<sup>n</sup> h*

 1

kernel introduced in (1). When the system input is a multi-tone function described by

1

*i*

integer), the system output frequency response can be evaluated as [9]:

*k kn*

 

*a*

*a* 

*<sup>k</sup>* can be explicitly written as sig( )

<sup>1</sup> <sup>1</sup>

sgn( ) 1 0

1

*a*

*CE H j j C C CE H*

 

with terminating condition 1)(1 *jHCE*

 

 

*N*

*n*

stead of the form in [9], 1 0

where )( *<sup>i</sup> F* 

polynomial form as

where <sup>1</sup> ( ,, ) *CE H j j n n* 

GFRF 1 ( ,, ) *Hj j n n* 

*K*

 

(2 ) *<sup>n</sup>*

where ),,( *<sup>n</sup>* <sup>1</sup> *<sup>n</sup> h* is the *n*th-order Volterra kernel which is a real valued function of *n* ,, <sup>1</sup> . For the same class of nonlinear systems, it can also be modelled by the following nonlinear differential equation (NDE)

$$\sum\_{m=1}^{M} \sum\_{p=0}^{m} \sum\_{k\_1, k\_m=0}^{K} c\_{p, m-p}(k\_1, \cdots, k\_m) \prod\_{i=1}^{p} \frac{d^{k\_i} y(t)}{dt^{k\_i}} \prod\_{i=p+1}^{m} \frac{d^{k\_i} u(t)}{dt^{k\_i}} = 0 \tag{2}$$

where 0 ( ) ( ) *k k k dxt x t dt* , *K k K k K kk qp qp* 00, 0 )()()( 1 1 , *M* is the maximum degree of nonlinearity

in terms of *y*(t) and *u*(t), and *K* is the maximum order of the derivative. In this model, the parameters such as *c*0,1(.) and *c*1,0(.) are referred to as linear parameters corresponding to coefficients of linear terms in the model, *i.e.*, ( ) *<sup>k</sup> k d yt dt* and ( ) *<sup>k</sup> k dut dt* for *k*=0,1,…,*K*; and , ( ) *p q c* for *p*+*q*>1 are referred to as nonlinear parameters corresponding to nonlinear terms in the model of the form 1 1 ( ) ( ) *<sup>i</sup> <sup>i</sup> i i p k p q k k k i ip d yt d ut dt dt* , *e.g.,* () () *<sup>p</sup> <sup>q</sup> <sup>y</sup> t ut* . The value *p*+*q* is referred to as the

nonlinear degree of parameter )( , *qp c* .

By using the probing method [16], a recursive algorithm for the computation of the *n*thorder 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]

$$Y(j\alpha) = \sum\_{n=1}^{N} \frac{1}{\sqrt{n} (2\pi)^{n-1}} \int\_{\alpha\_1 + \dots + \alpha\_n = \alpha} H\_n(j\alpha\_1, \dots, j\alpha\_n) \prod\_{i=1}^{n} \mathcal{U}(j\alpha\_i) d\sigma\_{\alpha} \tag{3}$$

which is truncated at the largest order *N* and where,

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

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]

> 1 1 1 ( ) (, ,) ( ) *N n*

<sup>1</sup> . For the same class of nonlinear systems, it can also be modelled by the following

*n i*

1 0, 0 1 1

*k*

*K*

in terms of *y*(t) and *u*(t), and *K* is the maximum order of the derivative. In this model, the parameters such as *c*0,1(.) and *c*1,0(.) are referred to as linear parameters corresponding to

*p*+*q*>1 are referred to as nonlinear parameters corresponding to nonlinear terms in the model

*k d yt dt*

*m p kk i ip*

*Mm K p k m k*

*n n ii*

is the *n*th-order Volterra kernel which is a real valued function of

( ) ( ) (, , ) <sup>0</sup> *<sup>i</sup> <sup>i</sup>*

(2)

 and ( ) *<sup>k</sup> k dut dt*

, *e.g.,* () () *<sup>p</sup> <sup>q</sup> <sup>y</sup> t ut* . The value *p*+*q* is referred to as the

 

(1)

*i i*

, *M* is the maximum degree of nonlinearity

for *k*=0,1,…,*K*; and , ( ) *p q c* for

*ut d*

*pm p m k k*

*d yt d ut c kk dt dt*

section which explains the main notations used in this paper is given in Appendix A.

**2. Frequency response functions of nonlinear systems** 

*y t h*

, 1

*K*

*kk qp qp* 00, 0 )()()(

*k*

where ),,( *<sup>n</sup>* <sup>1</sup> *<sup>n</sup> h* 

nonlinear differential equation (NDE)

0 ( ) ( ) 1

*K*

coefficients of linear terms in the model, *i.e.*, ( ) *<sup>k</sup>*

( ) ( ) *<sup>i</sup> <sup>i</sup> i i*

*d yt d ut dt dt*

*k k*

1 1

*i ip*

nonlinear degree of parameter )( , *qp c* .

*p k p q k*

*m*

,

1 1

*n*

*k k k dxt x t dt*

of the form

 ,, 

where

$$H\_n(jo\_1, \dots, jo\_n) = \int\_{-\infty}^{\infty} \cdots \int\_{-\infty}^{\infty} h\_n(\tau\_1, \dots, \tau\_n) \exp(-j(o\_1\tau\_1 + \dots + o\_n\tau\_n)) d\tau\_1 \cdots d\tau\_n \tag{4}$$

is known as the *n*th-order GFRF defined in [4], and ),,( *<sup>n</sup>* <sup>1</sup> *<sup>n</sup> h* is the *n*th-order Volterra kernel introduced in (1). When the system input is a multi-tone function described by

$$\mu(t) = \sum\_{i=1}^{\overline{K}} \left| F\_i \right| \cos(o\_i t + \angle F\_i) \tag{5}$$

(where *Fi* is a complex number, *<sup>i</sup> F* is the argument, *Fi* is the modulus, and *K* is a positive integer), the system output frequency response can be evaluated as [9]:

$$Y(joo) = \sum\_{n=1}^{N} \frac{1}{2^n} \sum\_{o\_{k\_1} + \cdots + o\_{k\_n} = o} H\_n(joo\_{k\_1} \cdots \prime, joo\_{k\_n}) F(oo\_{k\_1}) \cdots F(oo\_{k\_n}) \tag{6}$$

where )( *<sup>i</sup> F <sup>k</sup>* can be explicitly written as sig( ) ( ) *<sup>k</sup> <sup>i</sup> <sup>i</sup> i i j F k <sup>k</sup> <sup>k</sup> F Fe* for *ki* ,,1 *K* in stead of the form in [9], 1 0 sgn( ) 1 0 *a a a* , and <sup>1</sup> , , *<sup>i</sup> <sup>k</sup> <sup>K</sup>* .

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 *n*th-order GFRF can then be expressed into a more straightforward polynomial form as

$$H\_n(jo\_{1'}, \cdots, jo\_n) = \text{CE}\{H\_n(jo\_{1'}, \cdots, jo\_n)\} \cdot f\_n(jo\_{1'}, \cdots, jo\_n) \tag{7}$$

where <sup>1</sup> ( ,, ) *CE H j j n n* is referred to as the parametric characteristic of the *n*th-order GFRF 1 ( ,, ) *Hj j n n* , which can be recursively determined as

$$\text{CE}(H\_n(jo\_1, \dots, jo\_n)) = \mathbb{C}\_{0,n} \oplus \left( \bigoplus\_{q=1}^{n-1} \bigoplus\_{p=1}^{n-q} \mathbb{C}\_{p,q} \otimes \text{CE}(H\_{n-q-p+1}(\cdot)) \right) \oplus \left( \bigoplus\_{p=2}^n \mathbb{C}\_{p,0} \otimes \text{CE}(H\_{n-p+1}(\cdot)) \right) \tag{8}$$

with terminating condition 1)(1 *jHCE <sup>i</sup>* . Note that *CE* is a new operator with two operations " " and " " defined in [6,7] (the definition of *CE* can be referred to Appendix

B and more detailed discussions in [22]), and *C* ,*qp* is a vector consisting of all the (*p+q*)th degree nonlinear parameters, i.e.,

$$\mathbf{C}\_{p,q} = [\mathbf{c}\_{p,q}(0, \cdots, 0), \mathbf{c}\_{p,q}(0, \cdots, 1), \cdots, \mathbf{c}\_{p,q}(\underbrace{K, \cdots, K}\_{p+q=m})]$$

In Equation (8), 1 ( ,, ) *n n fj j* is a complex valued vector with the same dimension as <sup>1</sup> ( ,, ) *CE H j j n n* . In [7], a mapping function 1 ( ( ( )); , , ) *nn n CE H* from the parametric characteristic <sup>1</sup> ( ,, ) *CE H j j n n* to its corresponding correlative function ),,( *<sup>n</sup>* <sup>1</sup> *<sup>n</sup> jjf* is established as

 00 11 1 1 , () , , , (1) ( ( )) 1 , (1) ( ( )) 2 , (1) ( ( ) ) all the 2 partitions all the different for satisfying permut ( ) ( ) and 0 ( ( ) ( ) ( ); ) ( ( ), ( ); ) ( ( ( )); ) *k k p p q ns p q p q p q l lns p q l lns a x x pq l ln s q s ss c p cc c f c ns f s s sc* , x x 1 p , , all the p partitions for ( ) ations of {s , ,s } ( ( ( ))) , ( ( ) 1) ( ( ) ( ( ( )))) 1 ( ( ( )); ) *p q x pq i x pq i i s c p ns s c x pq lXi lXi ns s c i s sc* (9a)

where the terminating condition is *k*=0 and 1 1 (1; ) ( ) *i i H j* (which is the transfer function when all nonlinear parameters are zero), 1 {, }*<sup>p</sup> x x s s* is a permutation of 1 {, }*<sup>p</sup> x x s s* , *<sup>l</sup> snl* ))(()1( represents the frequency variables involved in the corresponding functions, *l*(*i*) for *i*=1… *sn* )( is a positive integer representing the index of the frequency variables, *x*

$$\overline{\mathbf{s}} = c\_{p\_0, q\_0}(\cdot) \mathbf{c}\_{p\_1, q\_1}(\cdot) \cdots c\_{p\_i, q\_i}(\cdot), \ n \langle \mathbf{s}\_x(\overline{\mathbf{s}}) \rangle = \sum\_{i=1} (p\_i + q\_i) - \mathbf{x} + 1 \,, \ \mathbf{x} \text{ is the number of the parameters in } \mathbf{x}$$
 
$$\underline{\mathbf{x}}$$

*<sup>x</sup> s* , 1 ( ) *i i i p q* is the sum of the subscripts of all the parameters in *<sup>x</sup> s* . Moreover,

$$\overline{X}(i) = \sum\_{j=1}^{i-1} m(s\_{\overline{x}\_j}(\overline{s}/c\_{p\eta}(\cdot)))\tag{9b}$$

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 7

*nn n CE H* 

(10)

 

1

*i*

*n*

(11a)

 

 

(11b)

 

 as

. Similarly, Equation

 

 

straightforward and meaningful polynomial function in terms of the first order GFRF and

11 1 ( , , ) ( , , ) ( () ; , , ) *H j j CE H j j CE H n n n nn n n*

 

 

<sup>1</sup>

*n nn*

 

() ( ,, ) ()

 

<sup>1</sup>

1 1

. Note that the

*nk k n*

 

2 2

 

 

> 

 

2 2

 

 

 

 

  can all be

0,3 1,1 0,2 1,1 1,1 2,0 2,1 1,2 2,0 0,2 2,0 3,0

*C CC CCC CCC C C C*

( ) ( , , ) () *<sup>n</sup>*

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

**Example 1**. Consider a simple example to demonstrate the results above. Suppose all the other nonlinear parameters in (2) are zero except *c*1,1(1,1), *c*0,2(1,1), *c*2,0(1,1). For convenience, *c*1,1(1,1) is written as *c*1,1 and so on. Consider the parametric characteristic of *H*3(.), which can

2 2

3 1 3 1,1 0,2 1,1 1,1 2,0 2,0 0,2 2,0 1,1 2,0 *CE H j j c c c c c c c c c c* ( ( , , )) [ , , , , , ]

1 1,1 1 3 2 1 1,1 0,2 1,1 1 2 2 1 0,2 1 2

 

( ( ),3; ) ( ( ( ) ( ) / ( )); , ) ( ( ( )); , )

 

31 3 2 1 2 3 1 32 1 2 ( ) ( ) ( ) ( ) ( )( ) *j j jjj j j j j Lj j Lj j Lj j Lj j*

3 1 2 123 1 2

1,1 0,2 1,1 1,1 2,0 2,0 0,2 2,0

Using (9abc), the correlative functions of each term in 31 3 *CE H j j* ( ,, )

( ) 1,1 0,2 (1) ( ( )) 3 1,1 0,2 1 3

( ( ) ( ); ) ( ( ) ( ); )

*f c f sc c c s c*

 

 

1 1,1 1 3 2 0,2 1 2 2 0,2 1 2

1 2

( ( ),3; ) ( ( ); , ) ( ( ); , )

obtained. For example, for the term *c*1,1*c*0,2, it can be derived directly from (9abc) that

*CC CCC C C C*

 

> *a a*

*f c f c c*

 

*c c c c*

 

*Y j CE H j j F j*

*Y j CE H j j F*

model parameters by using the mapping function 1 ( ( ( )); , , )

1

1 1

1

<sup>1</sup> ( ) ( ( ( )); , , ) ( ) ( ) <sup>2</sup> *n n*

 

*<sup>n</sup> <sup>n</sup> n nk k k k F j CE H F F*

*N*

*n*

*n n nn n i*

<sup>1</sup> ( ) ( ( ( ); , , ) ( )

*F j CE H Uj d*

*N*

*n*

Using (10), Equation (3) can be written as

*n*

1

*k kn*

 

(6) can be written as

easily be derived from (8),

*CE H j j*

31 3

( ,, )

Note that C1,1= *c*1,1, C0,2=*c*0,2, C2,0=*c*2,0. Thus,

*n s l lns*

 

 

 

*j*

  

where

where

1

 

convenient to be used for system analysis and design.

 

(2 ) *<sup>n</sup>*

$$L\_{\mathbb{A}}(j\varpi) = -\sum\_{k\_1=0}^{K} c\_{1,0}(k\_1)(j\varpi)^{k\_1} \qquad \forall \; \varpi \in \mathcal{R} \tag{9c}$$

$$\left(f\_1(\varepsilon\_{p,q}(\cdot), n(\overline{s}); \alpha\_{l(1)} \cdots \alpha\_{l(n(\overline{s}))})\right) = \left(\prod\_{i=1}^{q} (j\alpha\_{l(n(\overline{s}) - q + i)})^{k\_{p+i}}\right) \cdot \left(L\_{n(\overline{s})}(j\sum\_{i=1}^{n(\overline{s})} \alpha\_{l(i)}\right) \tag{9d}$$

$$f\_{\Delta a}(\mathbf{s}\_{\overline{\pi}\_1}\cdots\mathbf{s}\_{\overline{\pi}\_p}(\overline{\mathbf{s}}/\mathbf{c}\_{p,q}(\cdot)); o\_{l\{1\}}\cdots o\_{l\{n(\overline{\pi})-q\}}) = \prod\_{i=1}^p \left(j o\_{l\{\overline{\mathbf{X}}(i)+1\}} + \cdots + j o\_{l\{\overline{\mathbf{X}}(i)+n(\underline{\mathbf{s}\_{\overline{\pi}\_i}(\overline{\mathbf{x}}/\mathbf{c}\_{p\mathbf{\overline{\mathbf{s}}}(\cdot)}))\}}\right)^{k\_i} \tag{9e}$$

The mapping function <sup>1</sup> ( ( ( )); , , ) *nn n CE H* enables the complex valued function <sup>1</sup> ( ,, ) *n n fj j* to be analytically and directly determined in terms of the first order GFRF and nonlinear parameters. Therefore, the *n*th-order GFRF can directly be written into a more straightforward and meaningful polynomial function in terms of the first order GFRF and model parameters by using the mapping function 1 ( ( ( )); , , ) *nn n CE H* as

$$H\_n(jo\_1, \dots, jo\_n) = \text{CE}\left(H\_n(jo\_1, \dots, jo\_n)\right) \cdot \rho\_n(\text{CE}\left(H\_n(\cdot)\right); o\_1, \dots, o\_n) \tag{10}$$

Using (10), Equation (3) can be written as

$$Y(jao) = \sum\_{n=1}^{N} \text{CE}\left(H\_n(jao\_1, \dots, jo\_n)\right) \cdot \overline{F}\_n(jao) \tag{11a}$$

where 1 1 1 1 <sup>1</sup> ( ) ( ( ( ); , , ) ( ) (2 ) *<sup>n</sup> n n n nn n i i F j CE H Uj d n* . Similarly, Equation

(6) can be written as

6 Advances on Analysis and Control of Vibrations – Theory and Applications

all the 2 partitions all the different for satisfying permut

where the terminating condition is *k*=0 and 1 1 (1; ) ( )

*Ln*( *j*

The mapping function <sup>1</sup> ( ( ( )); , , ) 

*f c ns*

*s sc*

degree nonlinear parameters, i.e.,

In Equation (8), 1 ( ,, ) *n n*

 

() , , , (1) ( ( ))

*ns p q p q p q l lns*

( ( ) ( ) ( ); )

*k k*

00 11 ,, , () () () *k k pq pq pq sc c c* ,

( )

*p q*

*f s s sc*

<sup>1</sup> ( ,, ) *n n fj j* 

*i i*

 <sup>1</sup> ( ,, ) *CE H j j n n* 

),,( *<sup>n</sup>* <sup>1</sup> *<sup>n</sup> jjf*

( ) ( ) and 0

*p q*

 

1 ,

*s ss c p*

*<sup>l</sup>* 

*<sup>x</sup> s* ,

1

*i*

*x*

00 11

*cc c*

*fj j* 

parametric characteristic <sup>1</sup> ( ,, ) *CE H j j n n*

is established as

 

1

*i* 

when all nonlinear parameters are zero), 1

*p*

 

B and more detailed discussions in [22]), and *C* ,*qp* is a vector consisting of all the (*p+q*)th

,, , , [ (0, ,0), (0, ,1), , ( , , )] *pq pq p q p q*

*C c c cK K*

. In [7], a mapping function 1 ( ( ( )); , , )

1 , (1) ( ( )) 2 , (1) ( ( ) )

, ,

( ( ( ))) , ( ( ) 1) ( ( ) ( ( ( ))))

*x pq i x pq i i*

 *snl* ))(()1( represents the frequency variables involved in the corresponding functions, *l*(*i*) for *i*=1… *sn* )( is a positive integer representing the index of the frequency variables,

*ns s c x pq lXi lXi ns s c*

( ( ), ( ); ) ( ( ( )); )

*p q l lns a x x pq l ln s q*

x x 1 p

( ( ( )); )

 

> 

of {s , ,s }

 

*f c ns f s s sc*

all the p partitions for ( ) ations

 

,

*p q*

*s c*

1 ( ( )) ( ) 1 *x x ii i ns s p q x* 

is the sum of the subscripts of all the parameters in *<sup>x</sup> s* . Moreover,

1

*i*

*j Xi ns s c* 

) = 1

1,0 1 0

*c kj*

1 , (1) ( ( )) ( ( ) ) ( ) ()

 

 

and nonlinear parameters. Therefore, the *n*th-order GFRF can directly be written into a more

( ( ), ( ); )( ( ) ( ) *p i*

*p q l lns lns q i ns li*

 

(9e)

( )( )

1

<sup>1</sup> 2 , (1) ( ( ) ) ( ( ) 1) ( ( ) ( ( / ( )))) 1 ( ( ( )); ) ( ) *<sup>i</sup> p x pq i*

*a x x pq l lns q lXi lXi ns s c i*

> *nn n CE H*

*k*

*K*

1 ( ) ( ( ( ))) *<sup>j</sup>*

*x pq*

*k*

*pqm*

*nn n CE H* 

to its corresponding correlative function

*p*

{, }*<sup>p</sup> x x s s* is a permutation of 1

, *x* is the number of the parameters in

(9b)

enables the complex valued function

R (9c)

( )

1 1

*<sup>p</sup> <sup>k</sup>*

 

*i i*

 (9d)

*j j*

to be analytically and directly determined in terms of the first order GFRF

*<sup>q</sup> n s <sup>k</sup>*

*j L j*

1

 

 

*i i H j* (which is the transfer function

from the

(9a)

{, }*<sup>p</sup> x x s s* ,

is a complex valued vector with the same dimension as

$$Y(jao) = \sum\_{n=1}^{N} \text{CE} \left( H\_n(jo\_{k\_1}, \dots, jo\_{k\_n}) \right) \cdot \tilde{F}\_n(oo) \tag{11b}$$

where 1 1 1 <sup>1</sup> ( ) ( ( ( )); , , ) ( ) ( ) <sup>2</sup> *n n k kn <sup>n</sup> <sup>n</sup> n nk k k k F j CE H F F* . Note that the

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 *c*1,1(1,1), *c*0,2(1,1), *c*2,0(1,1). For convenience, *c*1,1(1,1) is written as *c*1,1 and so on. Consider the parametric characteristic of *H*3(.), which can easily be derived from (8),

$$\begin{split} & \mathbb{C}E\{H\_3(ja\_1, \dots, ja\_3)\} \\ &= \mathbb{C}\_{0,3} \oplus \mathbb{C}\_{1,1} \otimes \mathbb{C}\_{0,2} \oplus \mathbb{C}\_{1,1}^2 \oplus \mathbb{C}\_{1,1} \otimes \mathbb{C}\_{2,0} \oplus \mathbb{C}\_{2,1} \oplus \mathbb{C}\_{1,2} \oplus \mathbb{C}\_{2,0} \otimes \mathbb{C}\_{0,2} \oplus \mathbb{C}\_{2,0}^2 \oplus \mathbb{C}\_{3,0} \\ &= \mathbb{C}\_{1,1} \oplus \mathbb{C}\_{0,2} \oplus \mathbb{C}\_{1,1}^2 \oplus \mathbb{C}\_{1,1} \otimes \mathbb{C}\_{2,0} \oplus \mathbb{C}\_{2,0} \oplus \mathbb{C}\_{0,2} \oplus \mathbb{C}\_{2,0}^2 \end{split} $$

Note that C1,1= *c*1,1, C0,2=*c*0,2, C2,0=*c*2,0. Thus,

$$\text{CE}(H\_3(jo\_{1'}, \cdots, jo\_3)) = [c\_{1,1}c\_{0,2'}c\_{1,1'}^2c\_{1,1}c\_{2,0'}c\_{2,0}c\_{0,2'}c\_{2,0}c\_{1,1'}c\_{2,0}^2]^{-1}$$

Using (9abc), the correlative functions of each term in 31 3 *CE H j j* ( ,, ) can all be obtained. For example, for the term *c*1,1*c*0,2, it can be derived directly from (9abc) that

$$\begin{split} & \rho\_{\pi(\overline{\pi})} \{c\_{1,1}(\cdot)c\_{0,2}(\cdot); \alpha\_{0,1} \cdot \cdots \alpha\_{l(\pi(\overline{\pi}))} = \rho\_{\overline{\pi}} \{c\_{1,1}(\cdot)c\_{0,2}(\cdot); \alpha\_1 \cdots \alpha\_3\} \\ &= f\_1 \{c\_{1,1}(\cdot), \mathbf{3}; \alpha\_1 \cdots \alpha\_3\} \cdot f\_{2s} \{s\_1 \{c\_{1,1}(\cdot)c\_{0,2}(\cdot) \} / c\_{1,1}(\cdot)\}; \alpha\_1, \alpha\_2 \} \cdot \rho\_2 \{s\_1 \{c\_{0,2}(\cdot)\}; \alpha\_1, \alpha\_2\} \\ &= f\_1 \{c\_{1,1}(\cdot), \mathbf{3}; \alpha\_1 \cdots \alpha\_3\} \cdot f\_{2s} \{c\_{0,2}(\cdot); \alpha\_1, \alpha\_2\} \cdot \rho\_2 \{c\_{0,2}(\cdot); \alpha\_1, \alpha\_2\} \\ &= \frac{j\rho\_3}{L\_3(j\rho\_1 + \cdots + j\rho\_3)} \cdot \{j\rho\_1 + j\rho\_2\} \cdot \frac{j\rho\_1 j\rho\_2}{L\_2(j\rho\_1 + j\rho\_2)} = \frac{j\rho\_1 j\rho\_2/j\rho\_3 \{j\rho\_1 + j\rho\_2\}}{L\_3(j\rho\_1 + \cdots + j\rho\_3)L\_2(j\rho\_1 + j\rho\_2)} \end{split}$$

Proceed with the process above, the whole correlative function of 31 3 *CE H j j* ( ,, ) 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

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 9

. This will be discussed more in the next section. In this

 (12)

> 

)( . As demonstrated in

*nn n CE H* 

terms in the

 .

(13)

))(Im( are both

For any nonlinear parameter (simply denoted by *c*) in model (2), the output spectrum (11ab)

2 01 2 *Y j F j cF j c F j c F j* () () () () ()

Note that when *c* represents a nonlinearity from input terms, Equation (12) may be a finite

convergence. Thus these properties of this power series can be revealed by studying the

section, the alternating phenomenon of this power series and its influence are discussed.

sgn ( ) sgn (Re( )) sgn (Im( )) *cr r* 

**Definition 1** (Alternating series). Consider a power series of form (12) with c>0. If

2

 

Re( ( )) Re( ( )) Re( ( )) Re( ( ))

*Fj c Fj c Fj c F j j Fj c Fj c Fj c F j*

2

is an alternating series, then *jY*

alternating. When (12) is an alternating series, there are some interesting properties

(Im( ( )) Im( ( )) Im( ( )) Im( ( )) )

2 10 1 2 *Y j F j cF j c F j c F j* () () () () ()

 

*Fj Fj <sup>R</sup>*

 

> 

1 1

*Fj Fj*

 

 

Re( ( )) Im( ( )) min , Re( ( )) Im( ( )) *i i i i*

 

 

for i=0,1,2,3,…, then the series is an alternating series.

 

 

))(Re( and *jY*

(ℝ+) for *c*>0, then:

 

 

(14)

 

)(*i* for i=0,1,2,… are some scalar frequency functions and can be obtained

)(*i* dominates the fundamental properties of this power series such as

in (11a,b) by using the mapping function 1 ( ( ( )); , , )

 

series; in other cases, it is definitely an infinite series, and if only the first

can be expanded with respect to this parameter into a power series as

series (12) are considered, there is a truncation error denoted by

 

for *x* ℝ.

01 2

01 2

Example 1, *jF*

*r*

from ( ) *<sup>i</sup> F j*

For any

where

Clearly, *jF*

 or )( <sup>~</sup> *<sup>i</sup> jF* 

property of 1 ( ( ( )); , , ) 

sgn ( ) 0 0

<sup>1</sup> sgn ( ( )) sgn ( ( )) *c i c i Fj F j*

summarized in Theorem 1. Denote

From definition 1, if *Y j* ( )

*x x*

*nn n CE H* 

1 0

*x*

1 0

The series (12) can be written into two series as

*x*

( ) Re( ( )) (Im( ( )))

*Yj Yj j Yj*

 

(1) if there exist *T*>0 and *R*>0 such that for *i*>*T*

**Theorem 1**. Suppose (12) is an alternating series at a

ℂ, define an operator as

$$H\_3(j\alpha\_1,\cdots,j\alpha\_3) = [c\_{1,1}c\_{0,2}, c\_{1,1}^2, c\_{1,1}c\_{2,0}, c\_{2,0}c\_{0,2}, c\_{2,0}c\_{1,1}c\_{2,0}^2] \cdot \rho\_3(\text{CE}(H\_3(j\alpha\_1,\cdots,j\alpha\_3)); \alpha\_1,\cdots,\alpha\_3)$$

$$= c\_{1,1}c\_{0,2} \cdot \rho\_3(c\_{1,1}c\_{0,2}; \alpha\_1,\cdots,\alpha\_3) + c\_{1,1}^2 \cdot \rho\_3(c\_{1,1}^2; \alpha\_1,\cdots,\alpha\_3) + \dots + c\_{2,0}^2 \cdot \rho\_3(c\_{2,0}^2; \alpha\_1,\cdots,\alpha\_3)$$

As discussed in [7], it can be seen from Equations (10-11ab) and Example 1 that the mapping function 1 ( ( ( )); , , ) *nn n CE H* 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 1 (1) ( ) *H j<sup>l</sup>* 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 1 ( ( ( )); , , ) *nn n CE H* . Thus studying the properties of this power series is now equivalent to studying the properties of the mapping function 1 ( ( ( )); , , ) *nn n CE H* . Therefore, the mapping function <sup>1</sup> ( ( ( )); , , ) *nn n CE H* 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 1 ( ( ( )); , , ) *nn n CE H* 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.

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

$$Y(jao) = F\_0(jao) + cF\_1(jao) + c^2F\_2(jao) + \cdots + c^\rho F\_\rho(jao) + \cdots \tag{12}$$

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, *jF* )(*i* for i=0,1,2,… are some scalar frequency functions and can be obtained from ( ) *<sup>i</sup> F j* or )( <sup>~</sup> *<sup>i</sup> jF* in (11a,b) by using the mapping function 1 ( ( ( )); , , ) *nn n CE H* . Clearly, *jF* )(*i* 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 1 ( ( ( )); , , ) *nn n CE H* . 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

8 Advances on Analysis and Control of Vibrations – Theory and Applications

a polynomial form as (10-11ab), for example in this case

and 1 (1) ( ) *H j*

> 

mapping function 1 ( ( ( )); , , ) 

mapping function 1 ( ( ( )); , , ) 

nonlinear system to the system output spectrum.

 

<sup>1</sup> ( ( ( )); , , )

*nn n CE H* 

spectrum.

section.

Proceed with the process above, the whole correlative function of 31 3 *CE H j j* ( ,, )

2 2

*H j j c c c c c c c c c c CE H j j*

 

are explicitly determined by the mapping function 1 ( ( ( )); , , )

*nn n CE H* 

revealed by using the new mapping function 1 ( ( ( )); , , )

*cc cc c c c c*

*nn n CE H* 

 

1,1 0,2 3 1,1 0,2 1 3 1,1 3 1,1 1 3 2,0 3 2,0 1 3

 

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

3 1 3 1,1 0,2 1,1 1,1 2,0 2,0 0,2 2,0 1,1 2,0 3 3 1 3 1 3

As discussed in [7], it can be seen from Equations (10-11ab) and Example 1 that the

nonlinear systems such that the relationship between the frequency response functions and model parameters, and the relationship between the frequency response functions

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

the properties of this power series is now equivalent to studying the properties of the

for this frequency domain analysis to study the nonlinear influence on system output

In this study, a novel property of the nonlinear influence on system output spectrum is

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

**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

 

( , , ) [ , , , , , ] ( ( ( , , )); , , ) ( ; , , ) ( ; , , ) ... ( ; , , )

2 2 2 2

*<sup>l</sup>* can be demonstrated explicitly, and some new properties of the GFRFs

introduced above provides an important and significant technique

 

> 

 

can facilitate the frequency domain analysis of

*nn n CE H* 

*nn n CE H* 

. Therefore, the mapping function

 

 

. Thus studying

and frequency

 

 can

> 

$$\operatorname{sgn}\_c(\nu) = \begin{bmatrix} \operatorname{sgn}\_r(\operatorname{Re}(\nu)) & \operatorname{sgn}\_r(\operatorname{Im}(\nu)) \end{bmatrix}$$

where 1 0 sgn ( ) 0 0 1 0 *r x x x x* for *x* ℝ.

**Definition 1** (Alternating series). Consider a power series of form (12) with c>0. If <sup>1</sup> sgn ( ( )) sgn ( ( )) *c i c i Fj F j* for i=0,1,2,3,…, then the series is an alternating series.

The series (12) can be written into two series as

$$\begin{aligned} Y(jo) &= \text{Re}(Y(jo)) + j(\text{Im}(Y(jo))) \\ &= \text{Re}(F\_0(jo)) + c \, \text{Re}(F\_1(jo)) + c^2 \, \text{Re}(F\_2(jo)) + \cdots + c^\rho \, \text{Re}(F\_\rho(jo)) + \cdots \\ &+ j(\text{Im}(F\_0(jo)) + c \, \text{Im}(F\_1(jo)) + c^2 \, \text{Im}(F\_2(jo)) + \cdots + c^\rho \, \text{Im}(F\_\rho(jo)) + \cdots) \end{aligned} \tag{13}$$

From definition 1, if *Y j* ( ) is an alternating series, then *jY* ))(Re( and *jY* ))(Im( are both alternating. When (12) is an alternating series, there are some interesting properties summarized in Theorem 1. Denote

$$\text{If } Y(jo)\_{1 \to \rho} = F\_0(jo) + cF\_1(jo) + c^2F\_2(jo) + \dots + c^\rho F\_\rho(jo) \tag{14}$$

**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*

$$\min\left\{-\frac{\text{Re}(F\_i(jo))}{\text{Re}(F\_{i+1}(jo))}, -\frac{\text{Im}(F\_i(jo))}{\text{Im}(F\_{i+1}(jo))}\right\} > R$$

then (12) has a radius of convergence *R*, the truncation error for a finite order >*T* is 1 <sup>1</sup> () ( ) *cFj* , and for all *n* 0,

$$\left| Y(j\rho) \right| \in \Pi\_n = \left\| \left| Y(j\rho)\_{1 \to T + 2n + 1} \right|, \left| Y(j\rho)\_{1 \to T + 2n} \right| \right\| \text{and } \left| \prod\_{n+1} \subset \Pi\_n \right| ;$$

(2) <sup>2</sup> *Yj Yj Y j* ( ) ( )( ) is also an alternating series with respect to parameter *c*; Furthermore, <sup>2</sup> *Yj Yj Y j* ( ) ( )( ) is alternating only if Re( ( )) *Y j*is alternating;

(3) there exists a constant *<sup>c</sup>* <sup>0</sup> such that ( ) <sup>0</sup> *Y j c* for <sup>0</sup> *cc* .

*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<*c*<*R*, its truncation error and limit value can therefore be easily evaluated. The other two points of Theorem 1 imply that the magnitude of an alternating series can be suppressed by choosing a proper value for the parameter *c*. Therefore, once the system output spectrum can be expressed into an alternating series with respect to a model parameter (say *c*), it is easier to find a proper value for *c* such that the output spectrum is convergent, and the magnitude can be suppressed. Moreover, it is also shown that the lowest limit of the magnitude of the output spectrum that can be reached is larger than 1 1 ( ) *Y j <sup>T</sup>* and the truncation error of the output spectrum is less than the absolute value of the term of the largest order at the truncated point.

**Example 2**. Consider a single degree of freedom (SDOF) spring-damping system with a cubic nonlinear damping which can be described by the following differential equation

$$
\hbar m \ddot{y} = -k\_0 y - B\dot{y} - c\dot{y}^3 + \mu(t) \tag{15}
$$

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 11

 

, for 1, *<sup>l</sup> l kl k k*

 and 21 2 ( ,, )0 *Hj j n n* 

2 1

*n*

1 21

*n nn*

(19)

<sup>2</sup>*<sup>n</sup>* for n=0,1,2,3,… (16)

 

(17)

, and

is a

,1 1 <sup>1</sup> <sup>1</sup> ( , , ) ( , , )( ) *H j j Hj j j j n nn n*

 

, these results can be obtained directly as follows.

<sup>12</sup> *jjHCE*

*<sup>n</sup> FcjY* (18)

1 21

 

*n*

*n*

3 3 3

1 1 1

*i i i i i i i*

 

2 1

 

*k*

) *n*

3

1

*i i i i*

*i*

 

*j*

1 21

 

*CE H j F*

 

 

( ) ( )

*c* can be obtained

3 3 1 1

*L j L j*

( ) <sup>1</sup> ( (111); , , ) () () ( )

*c j Hj H j*

1

*<sup>n</sup>*

 

polynomial of )111( 0,3 *c* , and substituting these equations above into (11) gives another polynomial for the output spectrum. By using the relationship (10) and the mapping

*n*

Then the output spectrum at frequency can be computed as (*N* is the largest order after

2 1

*n*

<sup>1</sup> ( ) ( ( ( )); , , ) ( ) <sup>2</sup>

( ) ( ( ( )); , , <sup>2</sup>

 

0 <sup>12</sup> )( <sup>~</sup> )( *N*

2 1 2 1 21 21 12 2 1

*n n n kk d*

*n n n n kk d n*

*F j CE H jF k k k*

<sup>1</sup> ( ( ( )); , , ) ( 1) ( ) ( 1) <sup>2</sup>

*n nk*

21 21

3 3,0 1 2 3 3 3 1 1

 

Proceeding with the recursive computation above, it can be seen that 1 ( ,, ) *Hj j n n*

 

For simplicity, let ( ) sin( ) ( 0) *d d ut F t F* . Then ( )*<sup>l</sup> k ld F jk F*

21 1 21 21 21 1 21 ( , , ) ( () ; , , ) *<sup>n</sup> H j j c CE H n n nn n*

 

function <sup>1</sup> ( ( ( )); , , )

Therefore, for *n*=0,1,2,3,…

truncated)

where )( <sup>~</sup>

> 

*nn n CE H* 

*<sup>n</sup> <sup>n</sup> cjjHCE* ))1,1,1(()),,(( <sup>112</sup> 

*<sup>n</sup>*<sup>12</sup> *jF* can be computed as

1 21

 

1 21

 

*k k n*

*k k n*

1 21

 

*k k n*

2 1

according to equations (9a-c). For example,

*d n*

2 1 21 21

*<sup>F</sup> <sup>j</sup> CE H*

 

and 2 1 2 1 1 2 1 2 1 3,0 1 2 1 ( ( ) ; , , ) ( (1,1,1) ; , , ) *<sup>n</sup> n n nn CE H <sup>n</sup>*

*l n* 1, , in (11b). By using (8) or Proposition 5 in [6], it can be obtained that

*n*

<sup>12</sup> 0,3 and 0)),,(( *<sup>n</sup>*

Note that *k*0 represents the spring characteristic, *B* the damping characteristic and *c* is the cubic nonlinear damping characteristic. This system is a simple case of NDE model (2) and can be written into the form of NDE model with *M*=3, *K*=2, 1,0 *c m* (2) , 1,0 *c B* (1) , 0,1 <sup>0</sup> )0( *kc* , 3,0 *c c* (111) , 1)0( *c* 1,0 and all the other parameters are zero.

Note that there is only one nonlinear term in the output in this case, the *n*th-order GFRF for system (15) can be derived according to the algorithm in [1], which can be recursively determined as

$$\begin{aligned} H\_n(joo\_1, \cdots, joo\_n) &= \frac{c\_{3,0}(1, 1, 1)H\_{n,3}(joo\_1, \cdots, joo\_n)}{L\_n(joo\_1 + \cdots + jo\_n)} \\\\ H\_{n,3}(\cdot) &= \sum\_{i=1}^{n-2} H\_i(joo\_1, \cdots, joo\_i)H\_{n-i,2}(joo\_{i+1}, \cdots, ja\_n)(joo\_1 + \cdots + jo\_i) \end{aligned}$$

$$H\_{n,1}(jo\_1\cdots \prime, jo\_n) = H\_n(jo\_1\cdots \prime, jo\_n)(jo\_1 + \cdots + jo\_n)$$

Proceeding with the recursive computation above, it can be seen that 1 ( ,, ) *Hj j n n* is a polynomial of )111( 0,3 *c* , and substituting these equations above into (11) gives another polynomial for the output spectrum. By using the relationship (10) and the mapping function <sup>1</sup> ( ( ( )); , , ) *nn n CE H* , these results can be obtained directly as follows.

For simplicity, let ( ) sin( ) ( 0) *d d ut F t F* . Then ( )*<sup>l</sup> k ld F jk F* , for 1, *<sup>l</sup> l kl k k* , and *l n* 1, , in (11b). By using (8) or Proposition 5 in [6], it can be obtained that

$$CE(H\_{2n+1}(j\rho\_1,\cdots,j\rho\_{2n+1})) = (\mathcal{c}\_{1,0}(1,1,1))^n \text{ and } \ CE(H\_{2n}(j\rho\_1,\cdots,j\rho\_{2n})) = 0 \text{ for } n = 0, 1, 2, 3, \dots \tag{16}$$

Therefore, for *n*=0,1,2,3,…

10 Advances on Analysis and Control of Vibrations – Theory and Applications

*jY*

1 <sup>1</sup> () ( ) *cFj* 

 

(2) <sup>2</sup> *Yj Yj Y j* ( ) ( )( ) 

*Proof*. See Appendix C.□

1 1 ( ) *Y j*

are zero.

determined as

 

 

Furthermore, <sup>2</sup> *Yj Yj Y j* ( ) ( )( ) 

, and for all *n* 0,

 

 

(3) there exists a constant *<sup>c</sup>* <sup>0</sup> such that ( ) <sup>0</sup> *Y j*

of the term of the largest order at the truncated point.

1

2

*n*

*i*

1

*n n*

  

then (12) has a radius of convergence *R*, the truncation error for a finite order

*jY*

*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<*c*<*R*, its truncation error and limit value can therefore be easily evaluated. The other two points of Theorem 1 imply that the magnitude of an alternating series can be suppressed by choosing a proper value for the parameter *c*. Therefore, once the system output spectrum can be expressed into an alternating series with respect to a model parameter (say *c*), it is easier to find a proper value for *c* such that the output spectrum is convergent, and the magnitude can be suppressed. Moreover, it is also shown that the lowest limit of the magnitude of the output spectrum that can be reached is larger than

*<sup>T</sup>* and the truncation error of the output spectrum is less than the absolute value

3

3,0 ,3 1

 

(1,1,1) ( , , ) ( ,, ) ( )

( ) ( , , ) ( , , )( )

 

*n i i ni i n i*

*H Hj j H j j j j*

*c Hj j Hj j Lj j*

,3 1 ,2 1 1

1

*n n*

*n n*

   

> 

<sup>0</sup> *my k y By cy u t* ( ) (15)

**Example 2**. Consider a single degree of freedom (SDOF) spring-damping system with a cubic nonlinear damping which can be described by the following differential equation

Note that *k*0 represents the spring characteristic, *B* the damping characteristic and *c* is the cubic nonlinear damping characteristic. This system is a simple case of NDE model (2) and can be written into the form of NDE model with *M*=3, *K*=2, 1,0 *c m* (2) , 1,0 *c B* (1) , 0,1 <sup>0</sup> )0( *kc* , 3,0 *c c* (111) , 1)0( *c* 1,0 and all the other parameters

Note that there is only one nonlinear term in the output in this case, the *n*th-order GFRF for system (15) can be derived according to the algorithm in [1], which can be recursively

is alternating only if Re( ( )) *Y j*

is also an alternating series with respect to parameter *c*;

for <sup>0</sup> *cc* .

and *n*<sup>1</sup> *<sup>n</sup>* ;

is alternating;

])(,)([)( *<sup>n</sup> nT* <sup>121</sup> <sup>21</sup> *nT jY*

  >*T* is

$$H\_{2n+1}(\mathbf{j}a\_1\cdots\mathbf{j}a\_{2n+1}) = \mathbf{c}^n \cdot \rho\_{2n+1}(\mathbf{C}\mathbf{E}\{H\_{2n+1}(\cdot)\}; a\_1\cdots, a\_{2n+1}) \text{ and } H\_{2n}(\mathbf{j}a\_1\cdots\mathbf{j}a\_{2n}) = 0 \tag{17}$$

Then the output spectrum at frequency can be computed as (*N* is the largest order after truncated)

$$Y(j\Omega) = \sum\_{n=0}^{\left\lfloor \frac{N-1}{2} \right\rfloor} c^n \cdot \widetilde{F}\_{2n+1}(\Omega) + \dotsb \tag{18}$$

where )( <sup>~</sup> *<sup>n</sup>*<sup>12</sup> *jF* can be computed as

$$\begin{split} \tilde{F}\_{2n+1}(j\Omega) &= \frac{1}{2^{2n+1}} \sum\_{a\_{\mathbf{i}\_1} + \cdots + a\_{\mathbf{i}\_{2n+1}} = \Omega} \varphi\_{2n+1}(\operatorname{CE}(H\_{2n+1}(\cdot)); \boldsymbol{o}\_{\mathbf{i}\_1}, \cdots, \boldsymbol{o}\_{\mathbf{i}\_{2n+1}}) \cdot (-j\boldsymbol{F}\_d)^{2n+1} \cdot \boldsymbol{k}\_1 \boldsymbol{k}\_2 \cdots \boldsymbol{k}\_{2n+1} \\ &= \frac{1}{2^{2n+1}} \sum\_{a\_{\mathbf{i}\_1} + \cdots + a\_{\mathbf{i}\_{2n+1}} = \Omega} \varphi\_{2n+1}(\operatorname{CE}(H\_{2n+1}(\cdot)); \boldsymbol{o}\_{\mathbf{i}\_1}, \cdots, \boldsymbol{o}\_{\mathbf{i}\_{2n+1}}) \cdot (-1)^{n+1} j(\boldsymbol{F}\_d)^{2n+1} \cdot (-1)^n \\ &= -j \frac{\boldsymbol{F}\_d}{2} \}\_{a\_{\mathbf{i}\_1} + \cdots + a\_{\mathbf{i}\_{2n+1}} = \Omega} \varphi\_{2n+1}(\operatorname{CE}(H\_{2n+1}(\cdot)); \boldsymbol{o}\_{\mathbf{i}\_1}, \cdots, \boldsymbol{o}\_{\mathbf{i}\_{2n+1}}) \end{split} \tag{19}$$

and 2 1 2 1 1 2 1 2 1 3,0 1 2 1 ( ( ) ; , , ) ( (1,1,1) ; , , ) *<sup>n</sup> n n nn CE H <sup>n</sup> c* can be obtained according to equations (9a-c). For example,

$$\log\_3(c\_{3,0}(111); o\_1, o\_2, o\_3) = \frac{1}{L\_3(j\sum\_{i=1}^3 o\_i)} \cdot \prod\_{i=1}^3 (jo\_i) \cdot \prod\_{i=1}^3 H\_1(jo\_i) = \frac{\prod\_{i=1}^3 (jo\_i)}{L\_3(j\sum\_{i=1}^3 o\_i)} \cdot \prod\_{i=1}^3 H\_1(jo\_i)$$

1 3,0 , 5 3,0 3,0 1 5 1 3,0 1 5 2 3,0 1 5 all the 3 partitions all the different for (111) permutations of {0,0,1} ( ( ( ))) 3,0 ( () ( (111) (111); , , ) ( (111),5; , , ) ( ( (111)); ) ( ( (111)); *p x pq i i ax x c ns sc x lXi c c f c fs sc s c* , 3 1) ( ( ) ( ( ( )))) 1 2 0 0 1 3,0 1 5 1 1 1 2 3 3,0 3 5 1 3,0 1 5 2 0 1 0 3,0 1 5 1 1 3 3,0 2 4 1 5 2 100 ) ( ( (111)); ) (1; ) (1; ) ( (111); ) ( (111),5; , , ) ( ( (111)); ) (1; ) ( (111); ) (1; ) ( *x pq <sup>i</sup> lXi ns sc i a a a f sss c c f c f sss c c f sss* 3,0 1 5 3 3,0 1 3 1 4 1 5 ( (111)); ) ( (111); ) (1; ) (1; ) *c c* 

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 13

+(2.506378395908398e-010)c2-… (20b)

As pointed in Theorem 1, it is easy to find a *c* such that (20a-b) are convergent and their limits are decreased. From (20b) and according to Theorem 1, it can be computed that 0.01671739< *Y j* ( ) <0.0192276<0.0206882 for *c*=600. This can be verified by Figure 1. Figure 1 is a result from simulation tests, and shows that the magnitude of the output spectrum is decreasing when *c* is increasing. This property is of great significance in practical engineering systems for output suppression through structural characteristic design or

In this section, the conditions under which the output spectrum described by Equation (12) can be expressed into an alternating series with respect to any nonlinear parameter are studied. Suppose the system subjects to a harmonic input ( ) sin( ) ( 0) *d d ut F t F* and only the output nonlinearities (*i.e.*, *c*p,0(.) with *p* 2 ) are considered. For convenience, assume that there is only one nonlinear parameter *c*p,0(.) in model (2) and all the other nonlinear parameters are zero. The results for this case can be extended to the general one.

0 100 200 300 400 500 600 700 800 900 1000

c

Under the assumptions above, it can be obtained from the parametric characteristic analysis

in [6] as demonstrated in Example 2 and Equation (11b) that

feedback control.

**Figure 1.** Magnitude of output spectrum

**4. Alternating conditions** 

0.017

0.0175

0.018

0.0185

0.019

Magnitude of output spectrum

0.0195

0.02

0.0205

0.021

$$=\frac{1}{L\_5(j\sum\_{i=1}^5o\_i)}\cdot\left(\frac{(j\sum\_{i=3}^5o\_i)\prod\_{i=1}^5(jo\_i)}{L\_3(j\sum\_{i=3}^5o\_i)}+\frac{(j\sum\_{i=2}^5o\_i)\prod\_{i=1}^5(jo\_i)}{L\_3(j\sum\_{i=2}^5o\_i)}+\frac{(j\sum\_{i=1}^3o\_i)\prod\_{i=1}^5(jo\_i)}{L\_3(j\sum\_{i=1}^5o\_i)}\right)\cdot\prod\_{i=1}^5H\_1(jo\_i)$$

where {, } *<sup>i</sup>* , and so on. Substituting these results into Equations (18-19), the output spectrum is clearly a power series with respect to the parameter *c*. When there are more nonlinear terms, it is obvious that the computation process above can directly result in a straightforward multivariate power series with respect to these nonlinear parameters. To check the alternating phenomenon of the output spectrum, consider the following values for each linear parameter: *m*=240, *k*0=16000, *B*=296, *F*d=100, and 8.165 . Then it is obtained that

$$\begin{split} Y(j\Omega) &= \tilde{F}\_1(\Omega) + c\tilde{F}\_3(\Omega) + c^2 \tilde{F}\_5(\Omega) + \cdots \\ &= -j(\frac{F\_d}{2})H\_1(j\Omega) + 3(\frac{F\_d}{2})^3 \frac{\Omega^3 \left| H\_1(j\Omega) \right|^2 H\_1(j\Omega)}{L\_1(j\Omega)} \\ &+ 3(\frac{F\_d}{2})^5 \frac{\Omega^5 \left| H\_1(j\Omega) \right|^4 H\_1(j\Omega)}{L\_1(j\Omega)} (\frac{j6\Omega}{L\_1(j\Omega)} + \frac{j3\Omega}{L\_1(j\Omega)} + \frac{-j3\Omega}{L\_1(-j\Omega)}) + \cdots \\ &= (-0.02068817126756 + 0.000000114704116i) \end{split}$$

+(5.982851578532449e-006 -6.634300276113922e-010i)c

+(-5.192417616715994e-009 +3.323565122085705e-011i)c2+… (20a)

The series is alternating. In order to check the series further, computation of 2 1 3,0 1 2 1 ( (1,1,1) ; , , ) *<sup>n</sup> n n c* can be carried out for higher orders. It can also be verified that the magnitude square of the output spectrum (20a) is still an alternating series, *i.e.*,

$$\left| \begin{array}{c} \left| Y(j\Omega) \right|^2 = (4.280004317115985 \text{e-} 004) \text{-(2.475485177721052 \text{e-} 007) \text{c} } \right| \end{array} \right| $$

$$\star \text{(2.506378395908398e-010)} \text{c}^2 \dots \tag{20b}$$

As pointed in Theorem 1, it is easy to find a *c* such that (20a-b) are convergent and their limits are decreased. From (20b) and according to Theorem 1, it can be computed that 0.01671739< *Y j* ( ) <0.0192276<0.0206882 for *c*=600. This can be verified by Figure 1. Figure 1 is a result from simulation tests, and shows that the magnitude of the output spectrum is decreasing when *c* is increasing. This property is of great significance in practical engineering systems for output suppression through structural characteristic design or feedback control.

**Figure 1.** Magnitude of output spectrum

### **4. Alternating conditions**

12 Advances on Analysis and Control of Vibrations – Theory and Applications

 

5 3,0 3,0 1 5

*c c*

 ( (111) (111); , , )

where {, } 

that

 

  3,0

3

1

*i a a a* 2 100

*f c f sss c c f sss*

> 53 3 3 13 2 1

*Lj Lj L j L j*

2

() () ( ) ( ) 3( ) 2 2 ()

*F F Hj Hj j Hj L j*

4 5 5 1 1

13 5

*d d*

( ) () () ()

*Y j F cF c F*

1

*d*

2 1 3,0 1 2 1 ( (1,1,1) ; , , ) *<sup>n</sup> n n*

 

() () () ()

(

*c*

,

*x pq i i*

all the 3 partitions all the different for (111) permutations of {0,0,1}

*f c fs sc*

1 3,0 1 5 2 3,0 1 5

( (111),5; , , ) ( ( (111)); )

( ( ( ))) 3,0 ( ()

*f sss c c*

*ns sc x lXi*

*s c*

1 3,0 1 5 2 0 1 0 3,0 1 5 1 1 3 3,0 2 4 1 5

543 555

321 111 55 4 3 1

spectrum is clearly a power series with respect to the parameter *c*. When there are more nonlinear terms, it is obvious that the computation process above can directly result in a straightforward multivariate power series with respect to these nonlinear parameters. To check the alternating phenomenon of the output spectrum, consider the following values for each linear parameter: *m*=240, *k*0=16000, *B*=296, *F*d=100, and 8.165 . Then it is obtained

*j jj jj j*

 

*iii iii*

 

*ii i i ii i i*

( (111),5; , , ) ( ( (111)); ) (1; ) ( (111); ) (1; )

 

 

( ) ( )( ) ( )( ) ( ) <sup>1</sup> ( )

 

*<sup>i</sup>* , and so on. Substituting these results into Equations (18-19), the output

2 3 3 1 1

() () 63 3 3( ) ( ) 2 ( ) ( ) (3 ) ( )

=(-0.02068817126756 + 0.00000114704116i)

+(5.982851578532449e-006 -6.634300276113922e-010i)c

The series is alternating. In order to check the series further, computation of

<sup>2</sup> *Y j* ( ) = (4.280004317115985e-004)-(2.475485177721052e-007)c

the magnitude square of the output spectrum (20a) is still an alternating series, *i.e.*,

*c* can be carried out for higher orders. It can also be verified that

*F Hj Hj jj j*

1

 

1 11 1

*Lj Lj Lj L j*

+(-5.192417616715994e-009 +3.323565122085705e-011i)c2+… (20a)

*ii ii ii*

 

( ( (111));

1

 

2 0 0 1 3,0 1 5 1 1 1 2 3 3,0 3 5

 

 

( ( (111)); ) (1; ) (1; ) ( (111); )

 

> 

*ax x*

*p*

 

3,0 1 5 3 3,0 1 3 1 4 1 5 ( (111)); ) ( (111); ) (1; ) (1; ) *c c*

 

 

 

 

1) ( ( ) ( ( ( ))))

 

5

1

*H j*

*i*

*x pq <sup>i</sup> lXi ns sc*

,

 

*i*

)

 

 

 

> In this section, the conditions under which the output spectrum described by Equation (12) can be expressed into an alternating series with respect to any nonlinear parameter are studied. Suppose the system subjects to a harmonic input ( ) sin( ) ( 0) *d d ut F t F* and only the output nonlinearities (*i.e.*, *c*p,0(.) with *p* 2 ) are considered. For convenience, assume that there is only one nonlinear parameter *c*p,0(.) in model (2) and all the other nonlinear parameters are zero. The results for this case can be extended to the general one.

> Under the assumptions above, it can be obtained from the parametric characteristic analysis in [6] as demonstrated in Example 2 and Equation (11b) that

$$\begin{split} Y(j\Omega) &= Y\_1(j\Omega) + Y\_p(j\Omega) + \cdots + Y\_{(p-1)n+1}(j\Omega) + \cdots \\ &= \tilde{F}\_1(\Omega) + c\_{p,0}(\cdot)\tilde{F}\_p(\Omega) + \cdots + c\_{p,0}(\cdot)^n \tilde{F}\_{(p-1)n+1}(\Omega) + \cdots \end{split} \tag{21a}$$

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 15

, i.e.,

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*px p lXi lXi p x*

*i i*

*i i*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*i i*

*px p lXi lXi p x*

*i i*

*i*

*nn n* , *n*1+…+*n*e=*p*, *e* is the number of

(22)

*i*

**Theorem 2**. The output spectrum in (21a-c) is an alternating series with respect to parameter

 

( ) ' ( () ; ) ( )

\* ( ( ) 1) ( ( ) ( 1) 1)

(x , ,x ) ( ) (k , ,k )

<sup>1</sup> ( () ; ) ( )

1 2 ! (,,) !! ! *k p*

i

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

\* ( ( ) 1) ( ( ) ( 1) 1)

(x , ,x ) ( ) (k , ,k )

distinct differentials *k*i appearing in the combination, *n*i is the number of repetitions of *k*i, and

( 1) 1 ( 1) 1 () () Re( )Im( ) 0 () () *p n p n Hj Hj LjLj*

*c*

\*

*x*

*n*

 

*i*

*x*

1, 0 x n-1

 

*<sup>p</sup> nk k*

*lXi lXi p x*

*e*

, and

1 p

(x , ,x )

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1)

*px p lXi lXi p x*

*i i*

1)

)

*const*

(23)

 

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

*<sup>p</sup> <sup>x</sup>*

*lXi lXi p x*

 

*<sup>p</sup> <sup>x</sup>*

*c*p,0(k1,k2,…,kp) satisfying *cp*,0(.)>0 and *p r* 2 1 for r=1,2,3,...

1 ( 1) 1 ,0 (1) (( 1) 1) sgn ( 1) ( ( ) ; )

12 p 1p i

*H j <sup>c</sup> L j*

x x 1, 0 x n-1

 

*<sup>n</sup> j j <sup>n</sup>*

where *const* is a two-dimensional constant vector whose elements are +1, 0 or -1;

*n*

1 p

12 p 1 p

*<sup>n</sup> j j <sup>n</sup>*

1

<sup>1</sup> (,,) *x p nx x* .

all the different combinations of {x ,x ,...,x } satisfying x x 1, 0 x n-1

*p*

1

*i*

*n*

x x

*<sup>c</sup> Lj j*

1 p

1 p all the different 1 permutations of {k , ,k }

1 p all the different 1 permutations of {k , ,k }

*k i*

*n n <sup>c</sup> p n p l lp n c const*

1 ( 1) 1

1 ( 1) 1

\*

*k k p n*

 

( 1) 1 all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying

*p n i*

1 p

( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying

*k i*

(2) or if *k*1=*k*2=…=*k*p=*k* in *cp*,0(.), 1 1

12 p 1p i

( () ;

*pn l lp n i*

*k k p n*

 

 

( 1) 1 ,0 (1) (( 1) 1)

 

*n p n p l lp n*

*c*

( () ; )

\*

the termination is 1(1; ) 1 *<sup>i</sup>*

a similar definition holds for \*

1 ( 1) 1

*<sup>n</sup> <sup>c</sup>*

*k k p n*

 

sgn

 

 ; \*

1 p

1

sgn

*c*

*const*

(1) if and only if

where *<sup>i</sup> <sup>k</sup>* , ( 1) 1( ) *p n F j* can be computed from (11b), and *n* is a positive integer. Noting that ( )*<sup>l</sup> k ld F jk F* , 1, *<sup>l</sup> l kl k k* , and *l n* 1, , in (11b),

$$\begin{split} \tilde{F}\_{(p-1)n+1}(j\Omega) &= \frac{1}{2^{(p-1)n+1}} \\ \sum\_{a\_{k\_1} + \dots + a\_{k\_{(p-1)n+1}} = \Omega} \phi\_{(p-1)n+1}(c\_{p,0}(\cdot)^n; a\_{k\_1} \wedge \dots \wedge a\_{k\_{(p-1)n+1}}) \cdot (-jF\_d)^{(p-1)n+1} \cdot k\_1 k\_2 \cdots k\_{(p-1)n+1} \end{split} \tag{21b}$$

If *p* is an odd integer, then (*p*-1)*n*+1 is also an odd integer. Thus there should be (*p*-1)*n*/2 frequency variables being and (*p*-1)*n*/2+1 frequency variables being such that 1 ( 1) 1 *p n k k* . In this case,

$$(-jF\_d)^{(p-1)n+1} \cdot k\_1 k\_2 \cdots k\_{(p-1)n+1} = (-1) \cdot j \cdot \left( j^2 \right)^{(p-1)n/2} \cdot (F\_d)^{(p-1)n+1} \cdot (-1)^{(p-1)n/2} = -j(F\_d)^{(p-1)n+1}$$

If *p* is an even integer, then (*p*-1)*n*+1 is an odd integer for *n*=2*k* (*k*=1,2,3,…) and an even integer for *n*=2*k*-1 (*k*=1,2,3,…). When *<sup>n</sup>* is an odd integer, 1 ( 1) 1 *p n k k* for *<sup>l</sup> <sup>k</sup>* . This gives that ( 1) 1( ) *p n F j* =0. When *n* is an even integer, (*p*-1)*n*+1 is an odd integer. In this case, it is similar to that *p* is an odd integer. Therefore, for *n*>0

$$\widetilde{F}\_{(\rho-1)s+1}(j\Omega) = \begin{cases} -j\left(\frac{F\_d}{2}\right)^{(\rho-1)n+1} \sum\_{a\_{k\_1} + \dots + a\_{k\_{(\rho-1)s+1}} = \Omega} \sigma\_{(\rho-1)s+1}(\mathcal{c}\_{\rho,0}(\cdot)^{\*}; a\_{k\_1}, \dots, a\_{k\_{(\rho-1)s+1}}) & \text{if } p \text{ is odd or n is even} \\ 0 & \text{else} \end{cases} \tag{21c}$$

From Equations (21a-c) it is obvious that the property of the new mapping <sup>1</sup> ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n n pn p k k <sup>c</sup>* plays a key role in the series. To develop the alternating conditions for series (21a), the following results can be obtained.

**Lemma 1**. That 1 ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n n pn p k k <sup>c</sup>* is symmetric or asymmetric has no influence on ( 1) 1( ) *p n F j* .

Lemma 1 is obvious since 1 1)1( )( *<sup>k</sup> <sup>k</sup> np* includes all the possible permutations of

1 21 ( ,, ) *<sup>n</sup> k k* . Although there are many choices to obtain the asymmetric <sup>1</sup> ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n n pn p k k <sup>c</sup>* which may be different at different permutation 1 ( 1) 1 ( ,, ) *p n k k* , they have no effect on the analysis of )( <sup>~</sup> *np* 1)1( *jF* .

The following lemma is straightforward.

$$\begin{array}{llll}\textbf{Lemma} & \textbf{2.} & \textbf{For} & \boldsymbol{\nu}\_{1}\boldsymbol{\nu}\_{2}\boldsymbol{\nu}\in\mathbb{C}\_{\boldsymbol{\nu}} & \textbf{supp}pose & \textbf{sgn}\_{\boldsymbol{\varepsilon}}(\boldsymbol{\nu}\_{1})=-\textbf{sgn}\_{\boldsymbol{\varepsilon}}(\boldsymbol{\nu}\_{2}). & \textbf{If} & \textbf{Re}(\boldsymbol{\nu})\textbf{Im}(\boldsymbol{\nu})=\boldsymbol{0}, \quad \textbf{then} \\ \textbf{sgn}\_{\boldsymbol{\varepsilon}}(\boldsymbol{\nu}\_{1}\boldsymbol{\nu})=-\textbf{sgn}\_{\boldsymbol{\varepsilon}}(\boldsymbol{\nu}\_{2}\boldsymbol{\nu}). & \textbf{If} & \textbf{Re}(\boldsymbol{\nu})\textbf{Im}(\boldsymbol{\nu})=\textbf{0} \text{ and } \boldsymbol{\nu}\neq\boldsymbol{0}, \text{ then } \textbf{sgn}\_{\boldsymbol{\varepsilon}}(\boldsymbol{\nu}\_{1}\boldsymbol{\/\nu})=-\textbf{sgn}\_{\boldsymbol{\varepsilon}}(\boldsymbol{\nu}\_{2}\boldsymbol{\/\nu})\sqcap. \end{array}$$

**Theorem 2**. The output spectrum in (21a-c) is an alternating series with respect to parameter *c*p,0(k1,k2,…,kp) satisfying *cp*,0(.)>0 and *p r* 2 1 for r=1,2,3,...

(1) if and only if

14 Advances on Analysis and Control of Vibrations – Theory and Applications

, 1, *<sup>l</sup> l kl k k*

where *<sup>i</sup>* 

1 ( 1) 1 *p n*

*np*

on ( 1) 1( ) *p n F j* .

1 21 ( ,, ) *<sup>n</sup>*

1 ( 1) 1 ( ,, ) *p n*

**Lemma 2**. For 1 2

1 2 sgn ( ) sgn ( ) *c c*

 *k k*

> 

 *k k*

 *k k*

Noting that ( )*<sup>l</sup> k ld F jk F* 

*F j*

1 ( 1) 1

*k k p n*

 

This gives that ( 1) 1( ) *p n F j*

*<sup>j</sup> jF*

 

<sup>1</sup> ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n n pn p k k*

<sup>1</sup> ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n n pn p k k*

The following lemma is straightforward.

 , 

. If Re( )Im( ) 0

 

 

 

   

*d*

**Lemma 1**. That 1 ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n*

Lemma 1 is obvious since

1)1( *np <sup>k</sup> <sup>k</sup>*

1)1(

*np*

*<sup>k</sup>* , ( 1) 1( ) *p n F j*

( 1) 1 ( 1) 1

*p n p n*

. In this case,

<sup>1</sup> ( ) 2

1 ( 1) 1

*F cF c F*

() () () ()

*Yj Y j Y j Y j*

 

)(

1 1)1(

*<sup>k</sup> <sup>k</sup> np* 

, they have no effect on the analysis of )( <sup>~</sup> *np* 1)1( *jF* .

and

*pnp*

else 0

From Equations (21a-c) it is obvious that the property of the new mapping

*<sup>c</sup>* plays a key role in the series. To develop the alternating

. Although there are many choices to obtain the asymmetric

 0 , then 1 2 sgn ( ) sgn ( ) *c c* 

*<sup>c</sup>* which may be different at different permutation

, ℂ, suppose 1 2 sgn ( ) sgn ( ) *c c*

even isn or odd is p if),,;)(( )( <sup>2</sup> <sup>~</sup>

integer for *n*=2*k*-1 (*k*=1,2,3,…). When *<sup>n</sup>* is an odd integer, 1 ( 1) 1 *p n*

case, it is similar to that *p* is an odd integer. Therefore, for *n*>0

conditions for series (21a), the following results can be obtained.

 

*n pn p k k*

*<sup>c</sup> <sup>F</sup>*

1 1)1(

1 ,0 ,0 ( 1) 1

*p p n*

( ) () ( ) () ( )

1 ( 1) 1

( () ; , , ) ( ) *p n*

If *p* is an odd integer, then (*p*-1)*n*+1 is also an odd integer. Thus there should be (*p*-1)*n*/2 frequency variables being and (*p*-1)*n*/2+1 frequency variables being such that

If *p* is an even integer, then (*p*-1)*n*+1 is an odd integer for *n*=2*k* (*k*=1,2,3,…) and an even

( 1) /2 ( 1) 1 <sup>2</sup> ( 1) 1 ( 1) /2 ( 1) 1 1 2 ( 1) 1 ( ) ( 1) ( ) ( 1) ( ) *p n p n pn pn p n <sup>d</sup> p n <sup>d</sup> <sup>d</sup> jF k k k jj F j F*

*p p p pn*

, and *l n* 1, , in (11b),

( 1) 1 ,0 1 2 ( 1) 1

*c jF k k k*

*np k k*

*<sup>c</sup>* is symmetric or asymmetric has no influence

=0. When *n* is an even integer, (*p*-1)*n*+1 is an odd integer. In this

<sup>1</sup> 1)1( 0,1)1(

*n*

*n p n pn p k k d p n*

*n*

can be computed from (11b), and *n* is a positive integer.

( 1) 1

 *k k*

for *<sup>l</sup>*

(21c)

includes all the possible permutations of

 . If Re( )Im( ) 0 

 □.

, then

*<sup>k</sup>* .

(21a)

(21b)

1 ( 1) 1 1 ( 1) 1 ,0 (1) (( 1) 1) sgn ( 1) ( ( ) ; ) *k k p n n n <sup>c</sup> p n p l lp n c const* , i.e., 1 ( 1) 1 12 p 1p i 1 ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying x x 1, 0 x n-1 ( ) ' ( () ; ) ( ) sgn *i i i k k p n <sup>p</sup> <sup>x</sup> px p lXi lXi p x p n i n c H j <sup>c</sup> L j* 1 p \* 1 p \* ( ( ) 1) ( ( ) ( 1) 1) 1 p all the different 1 permutations of {k , ,k } (x , ,x ) ( ) (k , ,k ) *i i <sup>p</sup> <sup>x</sup> <sup>k</sup> lXi lXi p x k i <sup>n</sup> j j <sup>n</sup> const* (22)

where *const* is a two-dimensional constant vector whose elements are +1, 0 or -1;

$$=\frac{1}{L\_{(p-1)n+1}(j\alpha\_{\mathbb{Q}(1)}+\cdots+j\alpha\_{\mathbb{Q}((p-1)n+1)})}\cdot\sum\_{\substack{\mathbf{all}\text{ the different combinations}\\ \text{off }\{\mathbf{T}\_{1},\ldots,\mathbf{T}\_{p}\}\text{ satisfying}\\ \begin{subarray}{c}\prod\_{i=1}^{p}\mathsf{T}\_{i}\leftarrow\mathsf{T}\_{p}\end{subarray}}[\prod\_{i=1}^{p}\mathsf{q}\_{(p-1)\mathsf{T}\_{i}+1}^{\prime}\{\mathsf{c}\_{p,0}(\mathsf{t})^{\top}:o\_{\mathbb{Q}(\overline{\mathsf{X}}(i)+1)}\cdots o\_{\mathbb{Q}(\overline{\mathsf{X}}(i)+(p-1)\mathsf{T}\_{i}+1)}\}]}{\sum\_{\substack{\mathbf{l}\_{1},\ldots,\mathbf{l}\_{p}\in\mathbb{Z}\\ \mathbf{l}\_{1}+\cdots+\mathbf{l}\_{p}=n-1\end{subarray}}[\prod\_{i=1}^{p}\mathsf{q}\_{(p-1)\mathsf{T}\_{i}}^{\prime}\{\mathsf{c}\_{p,0}(\mathsf{t})^{\top}:o\_{\mathbb{Q}(\overline{\mathsf{X}}(i)+1)}\cdots o\_{\mathbb{Q}(\overline{\mathsf{X}}(i)+(p-1)\mathsf{T}\_{i}+1)}\}]}\\\cdot\frac{n\_{\mathrm{x}}^{\prime}\left(\overline{\mathsf{X}}\_{1}\cdots\overline{\mathsf{X}}\_{p}\right)}{n\_{\mathrm{x}}^{\prime}\left(\mathsf{k}\_{1}\cdots\mathsf{k}\_{p}\right)}\cdot\sum\_{\substack{\mathrm{all\text{the\'l\'ecir'}}\\ \text{all\text{the\'l\'ecir'}}\\ \text{ $\!\!\!\-per\mathrm{m}$ \!\!\-per\mathrm{m} $\!\!\-per\mathrm{m}$ \!\!\-per\mathrm{m} $\!\!\-per\mathrm{m}$ \!\!\-per$$

the termination is 1(1; ) 1 *<sup>i</sup>* ; \* 1 1 2 ! (,,) !! ! *k p e <sup>p</sup> nk k nn n* , *n*1+…+*n*e=*p*, *e* is the number of

distinct differentials *k*i appearing in the combination, *n*i is the number of repetitions of *k*i, and a similar definition holds for \* <sup>1</sup> (,,) *x p nx x* .

$$\text{(2) or if } k \coloneqq \mathbb{k} \equiv \dots \equiv k\_{\mathbb{P}} \coloneqq k \text{ in } c\_{\mathbb{P},0}(\text{.)}\\\text{, } \text{Re}(\frac{H\_1(j\Omega)}{L\_{(p-1)n+1}(j\Omega)}) \text{Im}(\frac{H\_1(j\Omega)}{L\_{(p-1)n+1}(j\Omega)}) = 0 \text{, and } \text{id}$$

$$\mathbf{sgn}\_{c}\begin{bmatrix}\sum\_{\substack{\boldsymbol{\alpha}\_{1}+\cdots+\boldsymbol{\alpha}\_{\boldsymbol{\beta}\_{\boldsymbol{\zeta}}}=1,\mathbf{0}\text{ all the different combinations}\\ \text{of }\{\overline{\mathbf{x}}\_{1},\overline{\mathbf{x}}\_{2},\cdots,\overline{\mathbf{x}}\_{p}\}\text{ satisfying}\\\overline{\mathbf{x}}\_{1}+\cdots+\overline{\mathbf{x}}\_{p}=\mathbf{n}-1,\mathbf{0}\text{ is}\mathbf{\overline{x}}\_{1}\leq \mathbf{n}-1\\\ \vdots \\\ \prod\_{i=1}^{p}\mathfrak{o}\_{(p-1)\overline{\mathbf{x}}\_{i}+1}^{p}\mathfrak{e}\_{p,0}^{\boldsymbol{\tau}}\{\boldsymbol{\up}\}^{\overline{\mathbf{x}}\_{i}};\operatorname{co}\_{l(\overline{\mathbf{x}}(i)+1)}\cdots\operatorname{co}\_{l(\overline{\mathbf{x}}(i)+(p-1)\overline{\mathbf{x}}\_{i}+1)\}\end{bmatrix}=\text{const}\tag{23}$$

$$\text{where if } \overline{\boldsymbol{\pi}}\_i = \boldsymbol{0}, \; \rho \boldsymbol{\sigma}\_{\{p-1\}\overline{\boldsymbol{\pi}}\_i + 1}^{\boldsymbol{\pi}} (\boldsymbol{c}\_{p,0} (\cdot)^{\overline{\boldsymbol{\pi}}\_i}; \boldsymbol{o}\_{l(\overline{\boldsymbol{\pi}}(i) + 1)} \cdots \boldsymbol{o}\_{l(\overline{\boldsymbol{\Delta}}(i) + (p-1)\overline{\boldsymbol{\pi}}\_i + 1)}) = 1 \text{, otherwise, } \overline{\boldsymbol{\pi}}\_i$$

$$\begin{split} &\rho^{\bullet}\_{(p-1)\overline{\pi}\_{i}+1}\{\mathcal{L}\_{p,0}\}^{\overline{\pi}\_{i}};o\_{l(\overline{\chi}(i)+1)}\cdots o\_{l(\overline{\chi}(i)+(p-1)\overline{\pi}\_{i}+1)}\} \\ &=\frac{\left(jo\_{l(\overline{\chi}(i)+1)}+\cdots+jo\_{l(\overline{\chi}(i)+(p-1)\overline{\pi}\_{i}+1)}\right)^{k}}{-\mathcal{L}\_{(p-1)\overline{\pi}\_{i}+1}\{jo\_{l(\overline{\chi}(i)+1)}+\cdots+jo\_{l(\overline{\chi}(i)+(p-1)\overline{\pi}\_{i}+1)\}}}. \\ &\sum\limits\_{\begin{subarray}{c}\mathsf{all the different combinations\\c\in\{\mathsf{x}\_{1},\mathsf{x}\_{2},\ldots,\mathsf{x}\_{p}\}\text{ satisfying}\\\mathsf{x}\_{1}+\cdots+\mathsf{x}\_{p}=\mathsf{x}\_{1}^{\mathsf{T}},1,0\le\mathsf{x}\_{i}\le 1\end{subarray}}n\_{\mathsf{x}}^{\ast}\left(\mathsf{x}\_{1},\ldots,\mathsf{x}\_{i}\right)\cdot\prod\limits\_{j=1}^{p}\phi^{\bullet}\_{(p-1)\overline{\pi}\_{j}+1}\{o\_{p,0}(\chi)^{\mathsf{x}\_{j}};o\_{l(\overline{\chi}(j)+1)},\cdots:o\_{l(\overline{\chi}(i)+(p-1)\chi\_{j}+1)}\}. \end{split}$$

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 17

*p n pq C c* can be computed as

, ,0

*nc s p pq p k k c sC c*

<sup>1</sup> ,0 ( ( )) ,0

*pq p*

*C c*

*n*

 

1 2 3 2 1 3,0 ( ( ) 1) ( ( ) 2 1)

2 1 3,0 ( ( ) 1) ( ( ) 2 1)

*x lX j lX j x*

*x*

( () ; ) *<sup>j</sup> j j*

*c*

*x x x lXi lXi x*

*i i*

(x ,x ,x ) ( ( ) ; ) *<sup>i</sup>*

*nn n CE H* 

 defined

,0

*nc s*

*d*

 

*F*

( ( ))

*n p*

(25a)

(25b)

*C c* (24b)

( ( )) ,0 , ,0 *<sup>n</sup>* ( ( \ ( )); ) *<sup>n</sup> <sup>p</sup> nc s <sup>p</sup> k k <sup>n</sup> nc s <sup>p</sup>*

( 1) 1 ( 1) 1 ( 1) 1 , p,0 ( ) ( ) ( ; \ (.)) *p n p n p n pq F F* 

*Cp* ,*<sup>q</sup>* includes all the nonlinear parameters in the system. Based on the parametric

in [7], (24b) can be determined easily. For example, suppose *p* is an odd integer larger than 1,

all the monomails consisting of the parameters in \ ( )

characteristic analysis in [6] and the new mapping function 1 ( ( ( )); , , )

i i

<sup>1</sup> ( ( )) ,0

 

*C c j*

parameter is greatly dependent on the system linear parameters.

all the different combinations 1

where, if *<sup>i</sup> x* =0, ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) 1 *<sup>i</sup>*

\*

*x*

*n*

  *i*

*x*

( () ; ) ( )

*n i*

atisfying x x x 1, 0 x n-1 *n*

123

2 1 3,0 ( ( ) 1) ( ( ) 2 1)

*i i*

*x lXi lXi x*

*j j Lj j*

( () ; ) ( )

*lXi lXi x x lXi lXi x*

all the different combinations of {x ,x ,x } satisfying x x x 1, 0 x

*i*

*x*

( ( ) 1) ( ( ) 2 1) 2 1 ( ( ) 1) ( ( ) 2 1)

 

 

*i i*

j

( )


*i*

*x*

123 i

of {x ,x ,x } s

*i*

*x*

*c*

123 123

2 1 3,0 (1) (2 1)

*n l ln*

 

*c*

(18-19). From Lemma 3 in Appendix D, it can be derived for this case that

2 1 1

 

*i i*

*k*

123

(x ,x ,x )

*px p lXi lXi p x*

1

*Lj j*

*<sup>n</sup> n k*

satisfying np (p q ) is odd and less than N ( ; \ (.)) <sup>2</sup>

where , ,0 ( \ ( )) *pq p sC c* denotes a monomial consisting of some parameters in , ,0 \ () *C c pq p* .

It is obvious that if (21a) is an alternating series, then (24a) can still be alternating under a proper design of the other nonlinear parameters (for example the other parameters are sufficiently small). Moreover, from the discussions above, it can be seen that whether the system output spectrum is an alternating series or not with respect to a specific nonlinear

**Example 3**. To demonstrate the theoretical results above, consider again model (15) in Example 2. Let ( ) sin( ) ( 0) *d d ut F t F* . The output spectrum at frequency is given in

> 2 1 (1) (2 1) <sup>3</sup> \*

*n c*

*c* , otherwise,

3

1

*j*

*n l l n*

( 1) [( ) ( )]

() 1 ()

 

*li li*

 

*j Hj*

*i*

then ( 1) 1( ) *p n F j* is given in (21c), and ( 1) 1 , p,0 ( ; \ (.))

( 1) 1 , p,0

*p n pq*

The recursive terminal of ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) *<sup>i</sup> i i x px p lXi lXi p x c* is *<sup>i</sup> x* =1.

*Proof*. See Appendix D. □

Theorem 2 provides a sufficient and necessary condition for the output spectrum series (21ac) to be an alternating series with respect to a nonlinear parameter *c*p,0(k1,k2,…,kp) satisfying *c*p,0(.)>0 and *p r* 2 1 for r=1,2,3,.... Similar results can also be established for any other nonlinear parameters. Regarding nonlinear parameter *c*p,0(k1,k2,…,kp) satisfying *c*p,0(.)>0 and *p r* 2 for r=1,2,3,...., it can be obtained from (21a-c) that

$$Y(j\Omega) = \tilde{F}\_1(\Omega) + c\_{p,0}(\cdot)^2 \tilde{F}\_{2(p-1)+1}(\Omega) + \dots + c\_{p,0}(\cdot)^{2n} \tilde{F}\_{2(p-1)n+1}(\Omega) + \dotsb$$

2( 1) 1( ) *p n F* for n=1,2,3,… should be alternating so that *Y j* ( ) is alternating. This yields that

$$\begin{split} & \text{sgn}\_{\boldsymbol{\varepsilon}} \left( \sum\_{o\_{b\_1} + \dots + o\_{b\_{2(p-1)n+1}} = \Omega} \rho\_{2(p-1)n+1} \{ \boldsymbol{c}\_{p,0} \boldsymbol{(\cdot)} \}^{2n}; o\_{l(1)} \cdots o\_{l(2(p-1)n+1)} \right) \\ & = -\text{sgn}\_{\boldsymbol{\varepsilon}} \left( \sum\_{o\_{b\_1} + \dots + o\_{b\_{2(p-1)(n+1)+1}} = \Omega} \rho\_{2(p-1)(n+1)+1} \{ \boldsymbol{c}\_{p,0} \boldsymbol{(\cdot)} \}^{2(n+1)}; o\_{l(1)} \cdots o\_{l(2(p-1)(n+1)+1)} \right) \end{split}$$

Clearly, this is different from the conditions in Theorem 2. It may be more difficult for the output spectrum to be alternating with respect to *c*p,0(.)>0 with *p r* 2 (even degree) than with respect to *c*p,0(.)>0 with *p r* 2 1 (odd degree).

Note that Equation (21a) is based on the assumption that there is only nonlinear parameter *c*p,0(.) and all the other nonlinear parameters are zero. If the effects from the other nonlinear parameters are considered, Equation (21a) can be written as

$$Y(j\Omega) = \tilde{F}\_1'(\Omega) + c\_{p,0}(\cdot)\tilde{F}\_p'(\Omega) + \dots + c\_{p,0}(\cdot)^n \tilde{F}\_{(p-1)n+1}'(\Omega) + \dotsb \tag{24a}$$

where

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 17

*n*

$$
\tilde{F}'\_{(p-1)n+1}(\Omega) = \tilde{F}\_{(p-1)n+1}(\Omega) + \delta\_{(p-1)n+1} \{ \Omega; C\_{p',q'} \backslash \varepsilon\_{p,0}(.) \}\tag{24b}
$$

*Cp* ,*<sup>q</sup>* includes all the nonlinear parameters in the system. Based on the parametric characteristic analysis in [6] and the new mapping function 1 ( ( ( )); , , ) *nn n CE H* defined in [7], (24b) can be determined easily. For example, suppose *p* is an odd integer larger than 1, then ( 1) 1( ) *p n F j* is given in (21c), and ( 1) 1 , p,0 ( ; \ (.)) *p n pq C c* can be computed as

16 Advances on Analysis and Control of Vibrations – Theory and Applications

where if *<sup>i</sup> x* =0, ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) 1 *<sup>i</sup>*

( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ( ( ) 1) ( ( ) ( 1) 1)

 

*lXi lXi p x p x lXi lXi p x*

*i i*

 

( () ; ) ( )

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*px p lXi lXi p x*

*i i*

fying

*p r* 2 for r=1,2,3,...., it can be obtained from (21a-c) that

1 2( 1)( 1) 1

parameters are considered, Equation (21a) can be written as

*k k p n*

 

*i i*

*x x*

1 2( 1) 1

respect to *c*p,0(.)>0 with *p r* 2 1 (odd degree).

*k k p n*

 

\*

The recursive terminal of ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) *<sup>i</sup>*

*x*

*n*

( )

 

*i*

*j j Lj j*

*x*

*c*

12 p

*Proof*. See Appendix D. □

2( 1) 1( ) *p n F*

where

1p j

all the different combinations of {x ,x ,...,x } satis

x x 1, 0 x -1

*x*

*i i*

1 p

*x*

*i*

(x , ,x )

*c* , otherwise,

*k*

1

*i i*

*c* is *<sup>i</sup> x* =1.

*px p lXi lXi p x*

Theorem 2 provides a sufficient and necessary condition for the output spectrum series (21ac) to be an alternating series with respect to a nonlinear parameter *c*p,0(k1,k2,…,kp) satisfying *c*p,0(.)>0 and *p r* 2 1 for r=1,2,3,.... Similar results can also be established for any other nonlinear parameters. Regarding nonlinear parameter *c*p,0(k1,k2,…,kp) satisfying *c*p,0(.)>0 and

> 2 2 1 ,0 2( 1) 1 ,0 2( 1) 1 ( ) ( ) () ( ) () ( ) *<sup>n</sup> Yj F c F p p p pn c F*

for n=1,2,3,… should be alternating so that *Y j* ( ) is alternating. This yields that

sgn ( () ; )

*c*

 

 

*c pn p l lp n*

Clearly, this is different from the conditions in Theorem 2. It may be more difficult for the output spectrum to be alternating with respect to *c*p,0(.)>0 with *p r* 2 (even degree) than with

Note that Equation (21a) is based on the assumption that there is only nonlinear parameter *c*p,0(.) and all the other nonlinear parameters are zero. If the effects from the other nonlinear

2( 1) 2( 1)( 1) 1 ,0 (1) (2( 1)( 1) 1)

 

 

*n*

1 ,0 ,0 ( 1) 1 ( ) ( ) () ( ) () ( ) *<sup>n</sup> Yj F c F c F p p p pn* (24a)

2 2( 1) 1 ,0 (1) (2( 1) 1)

*n*

sgn ( () ; )

*c*

 

*c pn p l lpn*

*j*

*<sup>p</sup> <sup>x</sup>*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*c*

*px p lX j lX j p x*

( () ; ) *<sup>j</sup> j j*

*px p lXi lXi p x*

,0 , ,0 i i ( ( )) ( 1) 1 , p,0 all the monomails consisting of the parameters in \ ( ) satisfying np (p q ) is odd and less than N ( ; \ (.)) <sup>2</sup> *p pq p nc s d p n pq C c F C c j* <sup>1</sup> ,0 ( ( )) ,0 <sup>1</sup> ( ( )) ,0 ( ( )) ,0 , ,0 *<sup>n</sup>* ( ( \ ( )); ) *<sup>n</sup> <sup>p</sup> nc s <sup>p</sup> k k <sup>n</sup> nc s <sup>p</sup> n nc s p pq p k k c sC c* 

where , ,0 ( \ ( )) *pq p sC c* denotes a monomial consisting of some parameters in , ,0 \ () *C c pq p* .

It is obvious that if (21a) is an alternating series, then (24a) can still be alternating under a proper design of the other nonlinear parameters (for example the other parameters are sufficiently small). Moreover, from the discussions above, it can be seen that whether the system output spectrum is an alternating series or not with respect to a specific nonlinear parameter is greatly dependent on the system linear parameters.

**Example 3**. To demonstrate the theoretical results above, consider again model (15) in Example 2. Let ( ) sin( ) ( 0) *d d ut F t F* . The output spectrum at frequency is given in (18-19). From Lemma 3 in Appendix D, it can be derived for this case that

$$\begin{split} \rho\_{2n+1}(\mathbf{c}\_{3,0}(\cdot)^{n}; o\_{l(1)} \cdots o\_{l(2n+1)}) &= \frac{(-1)^{n-1} \prod\_{i=1}^{2n+1} [(ja\_{l(i)})^{k} H\_{1}(ja\_{l(i)})]}{L\_{2n+1}(ja\_{l(1)} + \cdots + ja\_{l(2n+1)})} \\ &\cdot \sum\_{\substack{\text{all the different combinations} \\ \text{of } \{\underline{X}\_{1}, \underline{X}\_{2}\} \text{ satisfying} \\ \overline{\mathbf{x}}\_{1} + \overline{\mathbf{x}}\_{2} + \overline{\mathbf{x}}\_{3} = n-1, \ 0 \le \overline{\mathbf{x}}\_{1} \le n-1} \end{split} \tag{25a}$$

where, if *<sup>i</sup> x* =0, ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) 1 *<sup>i</sup> i i x px p lXi lXi p x c* , otherwise,

$$\begin{split} \rho\_{2\overline{\pi}\_{i}+1}^{\bullet}(c\_{3,0}\{\cdot\}^{\overline{\pi}\_{i}};o\_{l(\overline{\chi}(i)+1)}\cdots o\_{l(\overline{\chi}(i)+2\overline{\pi}\_{i}+1)})\\ =\frac{\left(jco\_{l(\overline{\chi}(i)+1)}+\cdots+jo\_{l(\overline{\chi}(i)+2\overline{\pi}\_{i}+1)}\right)^{k}}{-L\_{2\overline{\pi}\_{i}+1}(jco\_{l(\overline{\chi}(i)+1)}+\cdots+jo\_{l(\overline{\chi}(i)+2\overline{\pi}\_{i}+1)})}.\end{split}\tag{25b}$$
  $\sum\_{\begin{subarray}{c}\mathbf{n}\_{\chi}\text{``}\left(\mathbf{x}\_{1},\mathbf{x}\_{2},\mathbf{x}\_{3}\right)\cdots\prod\_{j=1}^{3}\mathbf{o}\_{2\mathbf{x}\_{j}+1}^{\bullet}\text{(\$\mathbf{c}\_{3,0}\$(\cdot)\$}\$^{\mathbf{x}\_{j}};o\_{l(\overline{\chi}(i)+1)}\cdots o\_{l(\overline{\chi}(i)+2\mathbf{x}\_{j}+1)}\$ or\left(\begin{subarray}{c}\mathbf{x}\_{1},\mathbf{x}\_{2},\mathbf{x}\_{3}\end{subarray};o\_{l(\overline{\chi}(i)+1)}\right)\right)}$ 

Note that the terminal condition for (25b) is at *<sup>i</sup> x* =1, i.e.,

$$\left. \left\langle \phi\_{2\overline{\pi}\_i + 1}^{\boldsymbol{\pi}} \{ \boldsymbol{c}\_{3,0} \} \right\rangle\_{\mathbb{R}}^{\overline{\pi}\_i}; o\_{\mathbb{I}\{ \overline{\boldsymbol{\pi}}(i) + 1 \}} \cdots o\_{\mathbb{I}\{ \overline{\boldsymbol{\pi}}(i) + 2 \overline{\pi}\_i + 1 \}} \right|\_{\overline{\pi}\_i = 1} = o\_{\mathbb{S}}'' \langle \boldsymbol{c}\_{3,0} \{ \cdot \}; o\_{\mathbb{I}\{ 1 \}} \cdots o\_{\mathbb{I}\{ 3 \}} \rangle = \frac{\left( \left\langle o\_{\mathbb{I}\{ 1 \}} + \cdots + \left\langle o\_{\mathbb{I}\{ 3 \}} \right\rangle \right)^k}{-L\_{\mathbb{S}} \left( \left\langle o\_{\mathbb{I}\{ 1 \}} + \cdots + \left\langle o\_{\mathbb{I}\{ 3 \}} \right\rangle \right)} \right) \tag{25c} \end{pmatrix}$$

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 19

(27a)

(27b)

2

  1

 

( )

 

( ) ( )

(27c)

1

( )

1

(28)

1

1

1

(27d)

1

*i*

(1) ( ) (1) ( )

(1) ( ) (1) ( )

*x l lx x*

2 1 0 ( ) 0

*x k jx B B j x km*

( ) 1 (1 ( ) ) ( ) (( ) )

*i i i x*

(1) ( )

 

*l l x*

*j j*

 

*l l x i i x l lx x i i i*

 

*j j j x j x*

 

*l l x*

 

(1) ( )

 

*j x*

*i*

*x* , it yields that

*i i i i*

> 

*k k n*

 

There always exists a combination such that

Note that (27b) holds for both 1)( *<sup>i</sup>*

Also noting that <sup>0</sup>*mkB* , all these show that

where ( ) { (2 1) 0 1 } *<sup>i</sup>*

If 0 *B km* , then it gives

n+1. Especially, when () 1 *<sup>i</sup>*

when () 1 *<sup>i</sup> x* ,

(a2), *i.e.*,

( ) ( ()) *i i i i*

*x j jn* , and *n* 1 denotes the odd integer not larger than

*l l x i x l lx x i j j j x L j j L jx*

( ) () *i i i i*

*j j j j L j j L j jB B*

( ) ( ()) ( ()) ( ())

( ) (( ) ) *i i i <sup>i</sup>*

*Lj j j x km*

*x l l x i x*

Note that in all the combinations involved in the summation operator in (26) or condition

1 2 n-1 1 21 i 1 2 n-1

<sup>1</sup> (1) ( )

*j j*

*<sup>n</sup> <sup>l</sup> l x*

<sup>1</sup> (1) ( )

*j j*

*<sup>n</sup> <sup>l</sup> l x*

1 (1) ( )

( )

 

*Lj j B*

<sup>1</sup> (1) ( ) <sup>1</sup> all the involved 1 (1) ( ) combinations <sup>1</sup> max ( ) ( )

*j j*

*<sup>n</sup> <sup>l</sup> l x*

*i i*

*<sup>n</sup> <sup>i</sup> x l l x*

 

*Lj j B*

 

 

*<sup>n</sup> <sup>i</sup> x l l x*

*i i*

1 (1) ( )

 

*L j j L jx mj x Bj x k*

(1) ( ) 0

(1) ( ) ( ) 0

all the combination (x ,x ,...,x ) satisfying x {2 1|1 1} x x .. x , and

" " happens only if x 2 2

( )

 

*Lj j B*

 

*i i*

*i*

 

*<sup>n</sup> <sup>i</sup> x l l x*

i 1

*i*

*x* , thus there is no combination such that

*i*

 

 

*j jn*

*i*

*x n*

Therefore, from (25a-c) it can be easily shown that 2 1 3,0 1 2 1 ( () ; ) *<sup>n</sup> n n c* can be written as

$$\begin{split} \phi\_{2n+1}(c\_{1,0}(\cdot)^{n}; o\_{1}\cdots o\_{2n+1}) \\ = \frac{(-1)^{n-1} \prod\_{i=1}^{2n+1} j o\_{i} H\_{1}(j o\_{i})}{L\_{2n+1}(j o\_{1} + \cdots + j o\_{2n+1})} \cdot \sum\_{\substack{\text{all the combination} \{\mathbf{x}\_{i}, \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n-1}\} \\ \text{ satisfying } \mathbf{x}\_{i}(2j+1) \mathbf{1} \le j \le n-1 \\ \mathbf{x}\_{i}^{\;\prime} \ge \mathbf{x}\_{i}^{\;\prime} \ge \mathbf{x}\_{n-1}^{\;\prime} \text{ and} \\ \mathbf{x}\_{i}^{\;\prime} \le \mathbf{x}\_{i}^{\;\prime} \text{ has property only } \mathbf{x}\_{i} + \mathbf{x}\_{i+1} \le 2n-2 \end{split}} \left( \frac{\mathbf{x}\_{i}^{\;\prime} \mathbf{x}\_{i} + \cdots + j o\_{l(\mathbf{x}\_{i})}}{L\_{2n}(j o\_{l(\mathbf{1})} + \cdots + j o\_{l(\mathbf{1}\_{k})})} \right) \tag{26}$$

where 1 2 n-1 (x ,x ,...,x ) *<sup>X</sup> r* is a positive integer which can be explicitly determined by (25ab) and represents the number of all the involved combinations which have the same <sup>1</sup> (1) ( ) <sup>1</sup> (1) ( ) ( ) *i i i <sup>n</sup> <sup>l</sup> l x i x l l x j j Lj j* . Therefore, according to the sufficient condition in Theorem 2, it

can be seen from (26) that the output spectrum (18) is an alternating series only if the following two conditions hold:

$$\begin{array}{ll} \text{(a1)} & \text{Re}(\frac{H\_{1}(j\Omega)}{L\_{2n+1}(j\Omega)})\text{Im}(\frac{H\_{1}(j\Omega)}{L\_{2n+1}(j\Omega)}) = 0 \\\\ \text{(a2)} & \text{sgn}\_{\varepsilon} \left( \sum\_{\begin{subarray}{c} \alpha\_{\mathbf{i}\_{1}} + \cdots + \alpha\_{\mathbf{j}\_{n-2}} = \alpha\_{\mathbf{i}} \text{2} \text{1} \text{ for combination} \left(\mathbf{x}\_{1}, \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n-1}\right) \\\\ \text{s.t.} & \text{as}\_{\mathbf{i}\_{1}} = \alpha\_{\mathbf{i}\_{2}} \text{2} \text{s} \text{ for combination} \left(\mathbf{x}\_{1}, \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n-1}\right) \end{subarray}}{\text{Call the combination}} \frac{r\_{\mathbf{X}}(\mathbf{x}\_{1}, \mathbf{x}\_{2}, \dots, \mathbf{x}\_{n-1})}{\sum\_{i=1}^{n} \left(\frac{\mathbf{j}\alpha\_{\mathbf{i}\_{1}}}{\mathbf{x}\_{i}} + \frac{\mathbf{j}\alpha\_{\mathbf{i}\_{1}}}{\mathbf{x}\_{i}} + \dots + \mathbf{j}\alpha\_{\mathbf{i}\_{\mathbf{i}\_{1}}}\right)} \right) = \text{const} \end{array} \tag{a}$$

Suppose <sup>0</sup> *<sup>k</sup> m* which is a natural resonance frequency of model (15). It can be derived that

$$L\_{2n+1}(j\Omega) = -\sum\_{k\_1=0}^{K} c\_{1,0}(k\_1)(j\Omega)^{k\_1} = -(m(j\Omega)^2 + B(j\Omega) + k\_0) = -jB\Omega$$

$$H\_1(j\Omega) = \frac{-1}{L\_1(j\Omega)} = \frac{1}{jB\Omega}$$

It is obvious that condition (a1) above is satisfied if 0 *<sup>k</sup> m* . Considering condition (a2), it can be derived that

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 19

$$\frac{j o\_{l(1)} + \dots + j o\_{l(x\_i)}}{-L\_{x\_i} (j o\_{l(1)} + \dots + j o\_{l(x\_i)})} = \frac{j \varepsilon(x\_i) \Omega}{-L\_{x\_i} (j \varepsilon(x\_i) \Omega)} \tag{27a}$$

where ( ) { (2 1) 0 1 } *<sup>i</sup> x j jn* , and *n* 1 denotes the odd integer not larger than n+1. Especially, when () 1 *<sup>i</sup> x* , it yields that

$$\frac{j o o\_{l(1)} + \dots + j o\_{l(x\_i)}}{-L\_{x\_i} (j o\_{l(1)} + \dots + j o\_{l(x\_i)})} = \frac{\pm j\Omega}{-L\_{x\_i} (\pm j\Omega)} = \frac{\pm j\Omega}{\pm jB\Omega} = \frac{1}{B} \tag{27b}$$

when () 1 *<sup>i</sup> x* ,

18 Advances on Analysis and Control of Vibrations – Theory and Applications

Note that the terminal condition for (25b) is at *<sup>i</sup> x* =1, i.e.,

*c c*

*i i i*

> 

> > *i*

 

2 1 2 1 () () Re( )Im( ) 0 () () *n n Hj Hj Lj Lj*

1 2 n-1 1 21 i 1 2 n-1

1

It is obvious that condition (a1) above is satisfied if 0 *<sup>k</sup>*

*k*

*K*

0

*c X*

satisfying x {2 1|1 1} x x .. x , and

" " happens only if x 2 2

i 1

*j jn*

1

 

*i*

*x n*

1

*k*

( ) ( )( ) ( ( ) ( ) )

*L j c k j m j B j k jB*

1 1 1 ( ) ( ) *H j L j jB* 

2 1 1,0 1 0

 

 

*i*

2 1 3,0 1 2 1 2 1 1

 

 

*n n n <sup>n</sup> <sup>n</sup>*

where 1 2 n-1 (x ,x ,...,x ) *<sup>X</sup>*

<sup>1</sup> (1) ( ) <sup>1</sup> (1) ( ) ( )

*j j Lj j*

*<sup>n</sup> <sup>l</sup> l x i x l l x*

(a2)

that

*i i*

following two conditions hold:

(a1) 1 1

Suppose <sup>0</sup> *<sup>k</sup>*

can be derived that

 

> 

*m*

*n*

*c*

( 1) ( )

( () ; )

2 1 3,0 ( ( ) 1) ( ( ) 2 1) 1 3 3,0 (1) (3)

i

x .. x , and " " happens only if x 2 2 *<sup>i</sup> x n*

2 n-1

satisfying x {2 1|1 1}

*j jn*

1

x

Therefore, from (25a-c) it can be easily shown that 2 1 3,0 1 2 1 ( () ; ) *<sup>n</sup>*

*x lXi lXi x x l l*

( ) ( () ; ) ( ( ); ) ( )

 

1 2 n-1

*jHj j j <sup>r</sup> Lj j Lj j*

i 1

21 1 21 all the combination (x ,x ,...,x ) 1 (1) ( )

represents the number of all the involved combinations which have the same

can be seen from (26) that the output spectrum (18) is an alternating series only if the

sgn (x ,x ,...,x ) ( )

*k k n i i*

which is a natural resonance frequency of model (15). It can be derived

*x l l*

<sup>1</sup> <sup>1</sup> (1) ( ) <sup>1</sup>

*r* is a positive integer which can be explicitly determined by (25ab) and

. Therefore, according to the sufficient condition in Theorem 2, it

1 2 n-1 all the combination (x ,x ,...,x ) 1 (1) ( )

2

*m*

(x ,x ,...,x ) ( ) ( )

*i i <sup>n</sup> <sup>l</sup> l x <sup>i</sup> X n n i x l l x*

 

(25c)

1 2 n-1

*n n*

 

(26)

(1) (3)

*j j*

<sup>1</sup> (1) ( )

 

. Considering condition (a2), it

*i x l l x*

*<sup>n</sup> <sup>l</sup> l x*

*j j <sup>r</sup> Lj j*

*i*

*const*

 

 

*Lj j*

*c* can be written as

*k*

3 (1) (3)

*i i*

*l l*

 

> 

> > *i*

 

 

$$\begin{split} \frac{jo\_{l(1)} + \dots + jo\_{l(x\_i)}}{-L\_{\mathbf{x}\_i} \left( jo\_{l(1)} + \dots + jo\_{l(\mathbf{x}\_i)} \right)} &= \frac{j\varepsilon \mathfrak{c}(\mathbf{x}\_i)\mathfrak{Q}}{-L\_{\mathbf{x}\_i} \left( j\varepsilon \mathfrak{c}(\mathbf{x}\_i)\mathfrak{Q} \right)} = \frac{j\varepsilon \mathfrak{c}(\mathbf{x}\_i)\mathfrak{Q}}{m \left( j\varepsilon \mathfrak{c}(\mathbf{x}\_i)\mathfrak{Q} \right)^2 + B \left( j\varepsilon \mathfrak{c}(\mathbf{x}\_i)\mathfrak{Q} \right) + k\_0} \\ = \frac{j\varepsilon \mathfrak{c}(\mathbf{x}\_i)\mathfrak{Q}}{(1 - \varepsilon \mathfrak{c}(\mathbf{x}\_i)^2)k\_0 + j\varepsilon \mathfrak{c}(\mathbf{x}\_i)\mathfrak{Q}\mathfrak{B}} = \frac{1}{B + j\left( \varepsilon \mathfrak{c}(\mathbf{x}\_i) - \frac{1}{\varepsilon \mathfrak{c}(\mathbf{x}\_i)} \right) \sqrt{k\_0m}} \end{split} \tag{27c}$$

If 0 *B km* , then it gives

$$\frac{j o\_{l(1)} + \dots + j o\_{l(x\_i)}}{-L\_{x\_i}(j o\_{l(1)} + \dots + j o\_{l(x\_i)})} \approx \frac{1}{j(\varepsilon(x\_i) - \frac{1}{\varepsilon(x\_i)})\sqrt{k\_0 m}}\tag{27d}$$

Note that in all the combinations involved in the summation operator in (26) or condition (a2), *i.e.*,

$$\sum\_{\substack{\alpha\_{k\_1}+\cdots+\alpha\_{k\_{2n+1}}=\Omega \text{ all the combination } (\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_{n-1})\\ \text{satisfying } \mathbf{x}\_i \in \{2j+1\} \text{ i.e.}\\ \stackrel{\mathbf{x}\_1 \ge \mathbf{x}\_2 \ge \dots \ge \mathbf{x}\_{n-1}, \text{ and}\\ \text{happyers only if } \mathbf{x}\_i + \mathbf{x}\_{i+1} \le 2n-2}} \text{(\-)}\text{)}$$

There always exists a combination such that

$$\prod\_{i=1}^{n-1} \frac{j o\_{l(1)} + \dots + j o\_{l(x\_i)}}{-L\_{\chi\_i}(j o\_{l(1)} + \dots + j o\_{l(x\_i)})} = \frac{1}{B^{n-1}} \tag{28}$$

Note that (27b) holds for both 1)( *<sup>i</sup> x* , thus there is no combination such that

$$\prod\_{i=1}^{n-1} \frac{j o o\_{l(1)} + \dots + j o\_{l(x\_i)}}{-L\_{x\_i} (j o\_{l(1)} + \dots + j o\_{l(x\_i)})} = -\frac{1}{B^{n-1}}$$

Also noting that <sup>0</sup>*mkB* , all these show that

$$\max\_{\substack{\text{all the involved} \\ \text{combinations}}} \left( \left| \prod\_{i=1}^{n-1} \frac{j o\_{l(1)} + \dots + j o\_{l(x\_i)}}{-L\_{\chi\_i} (j o\_{l(1)} + \dots + j o\_{l(x\_i)})} \right| \right) = \frac{1}{B^{n-1}}$$

which happens in the combination where (28) holds.

Because there are *n*+1 frequency variables to be and *n* frequency variables to be such that 1 21 *<sup>n</sup>* in (18-19), there are more combinations where () 0 *<sup>i</sup> x* that is 1 ( ) <sup>0</sup> (( ) ) *<sup>i</sup> <sup>i</sup> <sup>x</sup> x km* >0 in (27c-d). Thus there are more combinations where Im( (1) ( ) (1) ( ) ( ) *i i i l l x x l l x j j Lj j* ) is negative. Using (27b) and (27d), it can be shown under the Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 21

8.165, B=296<< <sup>0</sup> *k m* =1959.592. These are consistent with

the theoretical results established above. As it has been checked numerically in Example 2 that (18) is an alternating series, the theoretical results above are well verified by the real

Therefore, it can be seen that, at the driving frequency the system output spectrum (subject to a cubic nonlinear damping) can be designed to be an alternating series by properly designing system parameters (see conditions (b1-b2) above) and therefore can be suppressed as shown in Example 2 by properly choosing a value for the nonlinear parameter *c*. This result clearly demonstrate the mechanism for the nonlinear effect of the cubic nonlinear

More simulation studies about the properties of the cubic nonlinear damping can be referred to the simulation results in [8], where the effects of the cubic nonlinear damping are studied in details and compared with a linear damping. The case study here theoretically shows for the first time why and when these nonlinear effects happen and what the

Based on the discussions in Examples 2-3, it can be concluded that, the results of this study provide a new systematic method for the analysis and design of the nonlinear effect for a

Nonlinear influence on system output spectrum is investigated in this study from a novel perspective based on Volterra series expansion in the frequency domain. For a class of system nonlinearities, it is shown that system output spectrum can be expanded into an alternating series with respect to nonlinear parameters of the model under certain conditions and this alternating series has some interesting properties for engineering practices. Although there may be several existing methods such as perturbation analysis that can achieve similar objectives for some simple cases in practice, this study proposes a novel viewpoint on the nonlinear effect (i.e., alternating series) and on the analysis of nonlinear effect (i.e., the GFRFs-based) for a class of nonlinearities in the frequency domain. As some important properties of a linear system (e.g. stability) are determined by the positions of the poles of its transfer function, the fact of alternating series should be a natural characteristic of some important nonlinear effects for nonlinear systems in the frequency domain. This study provides some fundamental results for characterizing and understanding of nonlinear effects in the frequency domain from this novel viewpoint. The GFRFs-based analysis provides a useful technique for the analysis of nonlinear systems which is just similar to the transfer function based analysis for linear systems. The method demonstrated in this paper has been used for the analysis and design of nonlinear damping

(b2) The input frequency is <sup>0</sup> *<sup>k</sup>*

In Example 2, note that <sup>0</sup> *<sup>k</sup>*

damping in the frequency domain.

class of nonlinearities in the frequency domain.

underlying mechanism is.

**5. Conclusions** 

system.

*m* .

*m*

condition that 0 *B km* ,

$$\max\_{\substack{\text{all the involved} \\ \text{comditions}}} \left( \text{Im} (\prod\_{i=1}^{n-1} \frac{j o o\_{l(1)} + \dots + j o\_{l(x\_i)}}{-L\_{x\_i} (j o\_{l(1)} + \dots + j o\_{l(x\_i)})}) \right) \approx \frac{1}{B^{n-2} (\mathfrak{c}(x\_i) - \frac{1}{\mathfrak{c}(x\_i)}) \sqrt{k\_0 m}} \Big|\_{\mathfrak{c}(x\_i) = 3} = \frac{1}{2.7 B^{n-2} \sqrt{k\_0 m}}$$

This happens in the combinations where the argument of <sup>1</sup> (1) ( ) <sup>1</sup> (1) ( ) ( ) *i i i <sup>n</sup> <sup>l</sup> l x i x l l x j j Lj j* is either


$$\max\_{\substack{\mathbf{i}\in\mathbb{N}^{n}\text{ such}\\ \text{whose argument is}\\ -180^{\circ}}} \left| \mathrm{Im}(\prod\_{i=1}^{n-1} \frac{j a \mathbf{o}\_{l(\mathbf{l})} + \dots + j a \mathbf{o}\_{l(\mathbf{x}\_{i})}}{-L\_{x\_{i}} \left(j a \mathbf{o}\_{l(\mathbf{l})} + \dots + j a \mathbf{o}\_{l(\mathbf{x}\_{i})}\right)} \right| \approx \frac{1}{B^{n-4} \left(\boldsymbol{\varepsilon}(\mathbf{x}\_{i}) - \frac{1}{\boldsymbol{\varepsilon}(\mathbf{x}\_{i})}\right)^{3} \sqrt{k\_{0}m}} \Big|\_{\mathbf{c}^{\dagger}(\mathbf{x}\_{i}) = 3} = \frac{1}{2.7^{3}B^{n-4} \sqrt{k\_{0}m}} = \frac{1}{2.7^{3}B^{n-4}}$$

which is much less than *mkBn* 0 <sup>2</sup> 7.2 1 .

Therefore, if *B* is sufficiently smaller than *mk*<sup>0</sup> , the following two inequalities can hold for *n*>1

$$\begin{array}{l} \text{Re}\{\sum\_{\begin{subarray}{c}\{\mathbf{x}\_{1},\mathbf{x}\_{2},\ldots,\mathbf{x}\_{n-1}\end{subarray}\\ \text{as}\,\mathbf{i}\in[2j+1]\,\text{and}\,\mathbf{j}\neq\mathbf{i}\end{subarray}} r\_{\mathbf{X}}(\mathbf{x}\_{1},\mathbf{x}\_{2},\ldots,\mathbf{x}\_{n-1})\prod\_{i=1}^{n-1}\frac{j o\_{\{l(1)+\cdots+\cdots+j o\}\_{\{\mathbf{x}\_{i}\}}}}{L\_{\mathbf{x}\_{i}}(j o\_{l(1)}+\cdots+j o\_{l(\mathbf{x}\_{i})})})>0\\\quad\text{\$\mathbf{x}\_{1}\geq\mathbf{x}\_{2}\geq\mathbf{x}\_{n-1}\$, and}\\\text{Im}\{\sum\limits\_{\begin{subarray}{c}\mathbf{x}\_{1},\mathbf{x}\_{2},\ldots,\mathbf{x}\_{n-1}\\ \mathbf{x}\_{i}\in[2j+1]\,\mathbf{i}\leq j\leq n-1\end{subarray}}r\_{\mathbf{X}}(\mathbf{x}\_{1},\mathbf{x}\_{2},\ldots,\mathbf{x}\_{n-1})\prod\_{i=1}^{n-1}\frac{j o\_{l(1)}+\cdots+j o\_{l(\mathbf{x}\_{i})}}{L\_{\mathbf{x}\_{i}}(j o\_{l(1)}+\cdots+j o\_{l(\mathbf{x}\_{i})})})<0\\\quad\text{\$\mathbf{x}\_{1}\geq\mathbf{x}\_{2}\geq\mathbf{x}\_{n-1}\$, and}\\\quad\mathbf{x}\_{i}\geq\mathbf{x}\_{2}\geq\mathbf{x}\_{n-1}\$, and}r\_{\mathbf{x}}(\mathbf{x}\_{i},\mathbf{x}\_{j+1}\leq 2n-2) \end{array}$$

That is, condition (a2) holds for *n*>1 under 0 *B km* and <sup>0</sup> *<sup>k</sup> m* . Hence, (18) is an alternating series if the following two conditions hold:

(b1) *B* is sufficiently smaller than 0 *k m* ,

$$\text{(b2) The input frequency is } \Omega = \sqrt{\frac{k\_0}{m}} \text{.}$$

<sup>1</sup> (1) ( )

*j j*

This happens in the combinations where the argument of

<sup>1</sup> (1) ( )

*j j*

<sup>2</sup> 7.2

*<sup>n</sup> <sup>l</sup> l x*

*i i*

1 .

1 2 n-1

*x n*

1 2 n-1

*x n*

That is, condition (a2) holds for *n*>1 under 0 *B km* and <sup>0</sup> *<sup>k</sup>*

i 1

*i*

i 1

*j jn*

*j jn*

alternating series if the following two conditions hold:

*i*

*mkBn* 0

*<sup>n</sup> <sup>l</sup> l x*

Because there are *n*+1 frequency variables to be and *n* frequency variables to be

>0 in (27c-d). Thus there are more combinations where Im(

( )3 all the involved 2 2 <sup>1</sup> combinations <sup>1</sup> (1) ( ) ( ) 0 0 1 1 max ( Im( ) ) ( ) (( ) ) 2.7


3 3 ( )3 the combination <sup>4</sup> <sup>1</sup> <sup>3</sup> 3 4 <sup>1</sup> (1) ( ) whose argument is ( ) 0 0 -180 1 1 max ( Im( ) ) ( ) (( ) ) 2.7

Therefore, if *B* is sufficiently smaller than *mk*<sup>0</sup> , the following two inequalities can hold for *n*>1

1 2 n-1 all the combination (x ,x ,...,x ) <sup>1</sup> (1) ( ) satisfying x {2 1|1 1}

1 2 n-1 all the combination (x ,x ,...,x ) 1 (1) ( )

Im( (x ,x ,...,x ) ) 0 ( )

Re( (x ,x ,...,x ) ) 0 ( )

*<sup>x</sup> n n <sup>i</sup> x l l x <sup>i</sup> <sup>x</sup>*

 

*i i <sup>i</sup>*

is -1800, the absolute value of the corresponding imaginary part will be not more than

*i*

*i*

*X*

*X*

*r*

*r*

 

   

 

*<sup>i</sup> x l l x <sup>i</sup> <sup>x</sup>*

1 21 *<sup>n</sup>* in (18-19), there are more combinations where () 0 *<sup>i</sup>*

) is negative. Using (27b) and (27d), it can be shown under the

*Lj j B x km B km*

*i*

<sup>1</sup> (1) ( )

*j j*

*i x l l x*

*Lj j*

<sup>1</sup> (1) ( )

*j j*

*i x l l x*

*Lj j*

*<sup>n</sup> <sup>l</sup> l x*

*<sup>n</sup> <sup>l</sup> l x*

*i i*

*i i*

*Lj j B x km B km*

*i*

*i*

*i*

*i*

  . Hence, (18) is an

 

 

 

*m*

*i i*

*i*

is either

 

 

<sup>1</sup> (1) ( ) <sup>1</sup> (1) ( ) ( )

*j j Lj j*

*<sup>n</sup> <sup>l</sup> l x i x l l x*

*n n x*

*x* that is

which happens in the combination where (28) holds.

such that

0

 

(1) ( ) (1) ( ) ( )

 

*i i l l x x l l x*

condition that 0 *B km* ,

1 ( ) <sup>0</sup> (( ) ) *<sup>i</sup> <sup>i</sup> <sup>x</sup> x km* 

*j j Lj j*  

*i*

i 1 2 n-1

i 1 2 n-1

(b1) *B* is sufficiently smaller than 0 *k m* ,

satisfying x {2 1|1 1} x x .. x , and

" " happens only if x 2 2

" " happens only if x 2 2

x x .. x , and

which is much less than

  In Example 2, note that <sup>0</sup> *<sup>k</sup> m* 8.165, B=296<< <sup>0</sup> *k m* =1959.592. These are consistent with

the theoretical results established above. As it has been checked numerically in Example 2 that (18) is an alternating series, the theoretical results above are well verified by the real system.

Therefore, it can be seen that, at the driving frequency the system output spectrum (subject to a cubic nonlinear damping) can be designed to be an alternating series by properly designing system parameters (see conditions (b1-b2) above) and therefore can be suppressed as shown in Example 2 by properly choosing a value for the nonlinear parameter *c*. This result clearly demonstrate the mechanism for the nonlinear effect of the cubic nonlinear damping in the frequency domain.

More simulation studies about the properties of the cubic nonlinear damping can be referred to the simulation results in [8], where the effects of the cubic nonlinear damping are studied in details and compared with a linear damping. The case study here theoretically shows for the first time why and when these nonlinear effects happen and what the underlying mechanism is.

Based on the discussions in Examples 2-3, it can be concluded that, the results of this study provide a new systematic method for the analysis and design of the nonlinear effect for a class of nonlinearities in the frequency domain.

### **5. Conclusions**

Nonlinear influence on system output spectrum is investigated in this study from a novel perspective based on Volterra series expansion in the frequency domain. For a class of system nonlinearities, it is shown that system output spectrum can be expanded into an alternating series with respect to nonlinear parameters of the model under certain conditions and this alternating series has some interesting properties for engineering practices. Although there may be several existing methods such as perturbation analysis that can achieve similar objectives for some simple cases in practice, this study proposes a novel viewpoint on the nonlinear effect (i.e., alternating series) and on the analysis of nonlinear effect (i.e., the GFRFs-based) for a class of nonlinearities in the frequency domain. As some important properties of a linear system (e.g. stability) are determined by the positions of the poles of its transfer function, the fact of alternating series should be a natural characteristic of some important nonlinear effects for nonlinear systems in the frequency domain. This study provides some fundamental results for characterizing and understanding of nonlinear effects in the frequency domain from this novel viewpoint. The GFRFs-based analysis provides a useful technique for the analysis of nonlinear systems which is just similar to the transfer function based analysis for linear systems. The method demonstrated in this paper has been used for the analysis and design of nonlinear damping

systems. Further study will focus on more detailed design and analysis methods based on these results for practical systems.

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 23

*<sup>p</sup> xxx sss* <sup>21</sup> - A *p*-partition of a monomial 00 11 ,, , () () () *k k pq pq pq cc c*

0 *<sup>i</sup> x k* , and s0=1

: () () *nC f*

Consider a series

*f*1,*f*2,…, *f* 

where ℂ

is the

where the coefficients *ci* (*i*=1,…,

1. Reduced vectorized sum " ".

the elements in set (.).*C*<sup>2</sup>

2. Reduced Kronecker product " ".

3. Invariant. (a) ( ) () *CE H CE H* 

1 2 1 2 ( ) ( )( ) *CE H H CE H C CF CF CF F*

11 22 1 1 2 2

*CE H H CE H CE H C C VEC C i*

which implies that there are no repetitive elements in *C C* 1 2 .

*CF CF*

elements as those in C1.

following properties, also acting as operation rules:

*<sup>i</sup> <sup>x</sup> s* - A monomial of *x*i parameters of )}(,),({ , , <sup>00</sup> *qp qp kk c c* of the involved monomial,

 *Sn Sn* - A new mapping function from the parametric characteristics to the correlative functions, ( ) *CS n* is the set of all the monomials in the parametric characteristics

and ( ) *<sup>f</sup> S n* is the set of all the involved correlative functions in the nth order GFRF.

*H cf cf c f CF* 11 22

1 2 ( )[, , , ] *CE H c c c C CF*

parameters in a set*Cs* which takes values in ℂ, *fi* for *i*=1,…,*n* are some complex valued scalar functions in a set *Pf* which are independent of the parameters in*Cs* , denotes all the finite

 

ℂ

11 22 1 1 2 2 1 2 12 ( ) ( ) ( ) [,] *CE H H CE H CE H C C C C CF CF C F C F* , 2 21 2 *C VEC C C C* ( ) ,

where *C Ci i C C Ci i C* 1 1 12 2 2 ()1 , ()1 , *VEC*(.) is a vector consisting of all

12 3

[ (1) , , ( ) ] ( )()( ) ( ) <sup>1</sup> *CF CF C F C F*


) are different monomial functions in a set *<sup>c</sup> P* of some

is a vector including all the elements in C2 except the same

for this series such that

3 1 2 112

*C C C CCC*

3

*i C*

ℂ but is not a parameter of interest; (b)

*c* ], and *F*=[

( ( )) *<sup>x</sup> ns s* - The order of the GFRF where the monomial ( ) *xs s* is generated

**Appendix B: The Coefficient Extraction (CE) operator [6,7,22,23]** 

]T. Define a **Coefficient Extraction** operator *CE* : ℂ

order series with coefficients in *Pc* timing some functions in *<sup>f</sup> P* , *C*=[*c*1,*c*2,…,

### **Author details**

Xingiian Jing *Department of Mechanical Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong* 

## **Acknowledgement**

The author gratefully acknowledges the support of a GRF project of Hong Kong RGC (Ref 517810), the Department General Research Funds and Internal Competitive Research Grants of Hong Kong Polytechnic University for this work.

## **Appendix**

### **Appendix A: Nomenclature**

, 1 (,, ) *<sup>p</sup> q pq ck k* - A model parameter in the NDE model, *k*i is the order of the derivative, *p* represents the order of the involved output nonlinearity, *q* is the order of the involved input nonlinearity, and *p*+*q* is the nonlinear degree of the parameter.

<sup>1</sup> ( ,, ) *Hj j n n* - The nth-order GFRF

,, , , [ (0, ,0), (0, ,1), , ( , , )] *pq pq p q p q pqm C c c cK K* - A parameter vector consisting of all the

nonlinear parameters of the form ..

*CE*(.) - The coefficient extraction operator

<sup>1</sup> ( ( , , )) *CE H j j n n* - The parametric characteristics of the nth-order GFRF

<sup>1</sup> ( ,, ) *n n fj j* - The correlative function of 1 ( ( , , )) *CE H j j n n* 



(\*) ( ) and (\*) ( ) - The multiplication and addition by the reduced Kronecker product " "

and vectorized sum " " of the terms in (.) satisfying (\*), respectively

, , , <sup>1</sup> *k <sup>p</sup> q pq p q <sup>i</sup> CC C* - can be simply written as *<sup>k</sup> C* ,*qp* .

00 11 ,, , () () () *k k pq pq pq cc c* - A monomial consisting of nonlinear parameters

*<sup>p</sup> xxx sss* <sup>21</sup> - A *p*-partition of a monomial 00 11 ,, , () () () *k k pq pq pq cc c*

*<sup>i</sup> <sup>x</sup> s* - A monomial of *x*i parameters of )}(,),({ , , <sup>00</sup> *qp qp kk c c* of the involved monomial, 0 *<sup>i</sup> x k* , and s0=1

: () () *nC f Sn Sn* - A new mapping function from the parametric characteristics to the correlative functions, ( ) *CS n* is the set of all the monomials in the parametric characteristics and ( ) *<sup>f</sup> S n* is the set of all the involved correlative functions in the nth order GFRF.

( ( )) *<sup>x</sup> ns s* - The order of the GFRF where the monomial ( ) *xs s* is generated

### **Appendix B: The Coefficient Extraction (CE) operator [6,7,22,23]**

Consider a series

22 Advances on Analysis and Control of Vibrations – Theory and Applications

of Hong Kong Polytechnic University for this work.

nonlinearity, and *p*+*q* is the nonlinear degree of the parameter.


,, , , [ (0, ,0), (0, ,1), , ( , , )] *pq pq p q p q*

*C c c cK K*

*CE*(.) - The coefficient extraction operator

nonlinear parameters of the form ..

 

these results for practical systems.

**Author details** 

**Acknowledgement** 

**Appendix A: Nomenclature** 

Xingiian Jing

*Hong Kong* 

**Appendix** 

<sup>1</sup> ( ,, ) *Hj j n n* 

<sup>1</sup> ( ( , , )) *CE H j j n n* 

(\*)

, , , <sup>1</sup>

*<sup>p</sup> q pq p q <sup>i</sup> CC C*

<sup>1</sup> ( ,, ) *n n fj j* 

(\*) ( ) and

*k*

systems. Further study will focus on more detailed design and analysis methods based on

*Department of Mechanical Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon,* 

The author gratefully acknowledges the support of a GRF project of Hong Kong RGC (Ref 517810), the Department General Research Funds and Internal Competitive Research Grants

, 1 (,, ) *<sup>p</sup> q pq ck k* - A model parameter in the NDE model, *k*i is the order of the derivative, *p* represents the order of the involved output nonlinearity, *q* is the order of the involved input


( ) - The multiplication and addition by the reduced Kronecker product " "

 

*pqm*





and vectorized sum " " of the terms in (.) satisfying (\*), respectively

00 11 ,, , () () () *k k pq pq pq cc c* - A monomial consisting of nonlinear parameters


$$H\_{\rm CF} = c\_1 f\_1 + c\_2 f\_2 + \dots + c\_{\sigma} f\_{\sigma} \in \Xi$$

where the coefficients *ci* (*i*=1,…, ) are different monomial functions in a set *<sup>c</sup> P* of some parameters in a set*Cs* which takes values in ℂ, *fi* for *i*=1,…,*n* are some complex valued scalar functions in a set *Pf* which are independent of the parameters in*Cs* , denotes all the finite order series with coefficients in *Pc* timing some functions in *<sup>f</sup> P* , *C*=[*c*1,*c*2,…, *c* ], and *F*=[ *f*1,*f*2,…, *f* ]T. Define a **Coefficient Extraction** operator *CE* : ℂfor this series such that

$$\text{CE}(H\_{\text{CF}}) = [c\_{1'}c\_{2'}\cdots c\_{\sigma}] = \text{C} \in \text{C}^{\sigma}$$

where ℂ is the -dimensional complex valued vector space. This operator has the following properties, also acting as operation rules:

1. Reduced vectorized sum " ".

11 22 1 1 2 2 1 2 12 ( ) ( ) ( ) [,] *CE H H CE H CE H C C C C CF CF C F C F* , 2 21 2 *C VEC C C C* ( ) , where *C Ci i C C Ci i C* 1 1 12 2 2 ()1 , ()1 , *VEC*(.) is a vector consisting of all the elements in set (.).*C*<sup>2</sup> is a vector including all the elements in C2 except the same elements as those in C1.

2. Reduced Kronecker product " ".

$$\text{VCE}(H\_{\mathbb{C}\_1\mathbb{F}\_1}\cdot H\_{\mathbb{C}\_2\mathbb{F}\_2}) = \text{CE}(H\_{\mathbb{C}\_1\mathbb{F}\_1}) \otimes \text{CE}(H\_{\mathbb{C}\_2\mathbb{F}\_2}) = \text{C}\_1 \otimes \text{C}\_2 = V \text{EC} \left\{ \mathbb{C}\_3(i) \begin{vmatrix} \mathbb{C}\_3 = [\mathbb{C}\_1(1)\mathbb{C}\_2, \cdots, \mathbb{C}\_1(|\mathbb{C}\_1|)\mathbb{C}\_2] \\ 1 \le i \le \left| \mathbb{C}\_3 \right| \end{vmatrix} \right\}$$

which implies that there are no repetitive elements in *C C* 1 2 .

3. Invariant. (a) ( ) () *CE H CE H CF CF* ℂ but is not a parameter of interest; (b) 1 2 1 2 ( ) ( )( ) *CE H H CE H C CF CF CF F*

	- 4. Unitary. If *HCF* is not a function of *ci* for *i*=1…*n*, ( )1 *CE HCF* .

When there is a unitary 1 in *CE*(*HCF*), there is a nonzero constant term in the corresponding series *HCF* which has no relation with the coefficients *ci* (for *i*=1…*n*). In addition, if *HCF* =0, then CE(HCF)=0.

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 25

<sup>1</sup> ( ) Im( ( )) *<sup>I</sup> c Fj* 

are both alternating series and the absolute value of each

> 

 

1

*<sup>n</sup> <sup>n</sup>*

1,2,... 0

*n i*

 

is convergent. Similarly, it can be

 

are monotone decreasing, i.e.,

for *i*>*T*, then it can be

is convergent. According to

and Im( ( )) *Y j*

 

 

 

 

to be alternating.

 

for 0 *c c* . This

*i ni*

*c*

are

That is, lim Re( ( )) 0 *T n T n <sup>n</sup>*

 

proved that Im( ( )) *Y j*

bounded by

Since Re( ( )) *Y j*

<sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup> *i i i i cF j c F j*

2

of the terms in *Y j* ( )

Since Re( ( ) ( )) 0 1 *Fj F j* 

completes the proof. □

( ) ( )( )

 

*Yj Yj Y j*

term in Re( ( )) *Y j*

shown for Re( ( )) *Y j*

(2)

(3)

*cF j*

1 <sup>1</sup> ( ) Re( ( )) *<sup>R</sup> c Fj* 

Therefore, the truncation error for the series *Y j* ( )

and Im( ( )) *Y j*

 

and Im( ( )) *Y j*

 

 

0,1,2,... 0

( ) 1 ( ) 2( )

*Y j Y j c c Y j*

*n i*

*n n*

 

. Therefore, Re( ( )) *Y j*

is convergent. This proves that *Y j* ( )

[3], the truncation errors for the convergent alternating series Re( ( )) *Y j*

 

and Im( ( )) *Y j*

and <sup>1</sup>

 

and <sup>1</sup>

 

is

2 21 <sup>1</sup> () () () ( ) *R I cFj*

that for *n* 0

Re( ( ) ) Re( ( ) ) Re( ( )) Re( ( ) ) Re( ( ) ) *Y j*

1 1 1 21 1 2 <sup>1</sup> Im( ( ) ) Im( ( ) ) Im( ( )) Im( ( ) ) Im( ( ) ) *Y j*

01 2 0 1 2

It can be verified that the (2*k*)th terms in the series are positive and the (2*k*+1)th terms are negative for *k*=0,1,2,…. Moreover, it is not difficult to obtain that it needs only the real parts

*F j cF j c F j F j cF j c F j*

Therefore, 1 1 1 21 1 2 <sup>1</sup> ( ) () () () ( ) *Y j*

() ( )

2

 

0 1

  

 

 to be alternating for <sup>2</sup> *Yj Yj Y j* ( ) ( )( ) 

*i ni*

*c Fj F j*

*<sup>T</sup> Y j T n Yj Yj T n Y j <sup>T</sup>* .

( ( ) ( ) ( ) )( ( ) ( ) ( ) )

*<sup>T</sup> Y j T n Yj Yj T n Y j <sup>T</sup>*

1 1 *<sup>T</sup> Y j* 1 21 *T n Yj Yj* 1 2 *T n Y j* <sup>1</sup>*<sup>T</sup>*

<sup>1</sup> Im( ( )) Im( ( )) *i i i i cF j c F j*

2 2

 

<sup>1</sup> Re( ( ) ( )) () ( ) 2( )

*F j F j c nc F j F j Y j*

<0, there must exist *<sup>c</sup>* <sup>0</sup> such that ( ) <sup>0</sup> *Y j*

  - 5. Inverse. *CE-*1(*C*)=*HCF.*

### **Appendix C: Proof of theorem 1**

(1) *Y j* ( ) is convergent if and only if Re( ( )) *Y j* and Im( ( )) *Y j* are both convergent. Since *Y j* ( ) is an alternating series, Re( ( )) *Y j* and Im( ( )) *Y j* are both alternating from Definition 1. Then according to [3], *jY* ))(Re( is convergent if <sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup> *i i i i cF j c F j* and lim Re( ( )) 0 *<sup>i</sup> <sup>i</sup> <sup>i</sup> cF j* . Therefore, if there exists *T*>0 such that <sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup> *i i i i cF j c F j* for *i*>*T* and lim Re( ( )) 0 *<sup>i</sup> <sup>i</sup> <sup>i</sup> cF j* , the alternating series Re( ( )) *Y j* is also convergent. Now since there exist *T*>0 and *R*>0 such that 1 Re( ( )) Re( ( )) *i i F j <sup>R</sup> F j* for *i*>*T* and note *c*<*R*, it can be obtained that for *i*>*T*

$$-\frac{\operatorname{Re}(c^{i+1}F\_{i+1}(jo))}{\operatorname{Re}(c^{i}F\_{i}(jo))} = -\frac{\operatorname{Re}(cF\_{i+1}(jo))}{\operatorname{Re}(F\_{i}(jo))} = \left|\frac{\operatorname{Re}(cF\_{i+1}(jo))}{\operatorname{Re}(F\_{i}(jo))}\right| < \frac{c}{R} < 1$$

*i.e.*, <sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup> *i i i i cF j c F j* for *i*>*T* and *c*<*R*. Moreover, it can also be obtained that for *n*>0

$$\left| \text{Re}(F\_{T+n}(jo)) \right| < \frac{1}{R^n} \left| \text{Re}(F\_T(jo)) \right| $$

It further yields that

$$\left| \mathrm{Re}(\boldsymbol{c}^{T+n} F\_{T+n}(j\boldsymbol{o})) \right| < (\frac{c}{R})^n \boldsymbol{c}^T \left| \mathrm{Re}(F\_T(j\boldsymbol{o})) \right|.$$

That is, lim Re( ( )) 0 *T n T n <sup>n</sup> cF j* . Therefore, Re( ( )) *Y j* is convergent. Similarly, it can be proved that Im( ( )) *Y j* is convergent. This proves that *Y j* ( ) is convergent. According to [3], the truncation errors for the convergent alternating series Re( ( )) *Y j* and Im( ( )) *Y j* are bounded by

$$\left| o\_R(\rho) \right| \le c^{\rho+1} \left| \text{Re}(F\_{\rho+1}(j\rho)) \right| \text{ and } \left| o\_I(\rho) \right| \le c^{\rho+1} \left| \text{Im}(F\_{\rho+1}(j\rho)) \right| $$

Therefore, the truncation error for the series *Y j* ( ) is

24 Advances on Analysis and Control of Vibrations – Theory and Applications

addition, if *HCF* =0, then CE(HCF)=0.

same function series and are thus equivalent.

is convergent if and only if Re( ( )) *Y j*

**Appendix C: Proof of theorem 1** 

1. Then according to [3], *jY*

for *i*>*T* and lim Re( ( )) 0 *<sup>i</sup> <sup>i</sup> <sup>i</sup>*

*i.e.*, <sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup> *i i i i cF j c F j*

is an alternating series, Re( ( )) *Y j*

*cF j*

*i*

*i*

since there exist *T*>0 and *R*>0 such that

(1) *Y j* ( ) 

lim Re( ( )) 0 *<sup>i</sup> <sup>i</sup> <sup>i</sup>*

*cF j*

obtained that for *i*>*T*

It further yields that

*Y j* ( ) 

for *n*>0

5. Inverse. *CE-*1(*C*)=*HCF.* 6. 1 1 2 2

4. Unitary. If *HCF* is not a function of *ci* for *i*=1…*n*, ( )1 *CE HCF* .

When there is a unitary 1 in *CE*(*HCF*), there is a nonzero constant term in the corresponding series *HCF* which has no relation with the coefficients *ci* (for *i*=1…*n*). In

()( ) *CE H CE H C F C F* if the elements of *C*1 are the same as those of *C*2, where " " means equivalence, *i.e.*, both series are in fact the same result considering the order of *cifi* in the series has no effect on the value of a function series *HCF*. This further implies that the *CE* operator is also commutative and associative, for instance, 11 22 22 11 1 2 2 1 () () *CE H H C C CE H H C C CF CF CF CF* . Hence, the results by the CE operator with respect to the same purpose may be different but all correspond to the

7. Separable and interested parameters only. A parameter in a series can only be extracted if the parameter is interested and the series is separable with respect to this interested

and Im( ( )) *Y j*

, the alternating series Re( ( )) *Y j*

1 Re( ( )) Re( ( )) *i i*

*F j* 

 

<sup>1</sup> Re( ( )) Re( ( )) Re( ( )) 1 11 <sup>1</sup>

*c F j cF j cF j c cF j Fj Fj R*

<sup>1</sup> Re( ( )) Re( ( )) *T n <sup>n</sup> <sup>T</sup> F j Fj <sup>R</sup>* 

Re( ( )) ( ) Re( ( )) *T n n T T n T <sup>c</sup> c F j c Fj <sup>R</sup>* 

Re( ( )) Re( ( )) Re( ( ))

*i i i*

*i ii*

and Im( ( )) *Y j*

))(Re( is convergent if <sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup>

. Therefore, if there exists *T*>0 such that <sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup>

*F j <sup>R</sup>*

for *i*>*T* and *c*<*R*. Moreover, it can also be obtained that

are both convergent. Since

is also convergent. Now

 and

are both alternating from Definition

*i i i i cF j c F j*

*i i i i cF j c F j*

for *i*>*T* and note *c*<*R*, it can be

 

parameter. Thus the operation result is different for different purposes. �

$$\left| o(\rho) \right| = \sqrt{o\_{\mathbb{R}}(\rho)^2 + o\_{l}(\rho)^2} \le c^{\rho+1} \left| F\_{\rho+1}(j\rho o) \right| $$

Since Re( ( )) *Y j* and Im( ( )) *Y j* are both alternating series and the absolute value of each term in Re( ( )) *Y j* and Im( ( )) *Y j* are monotone decreasing, i.e., <sup>1</sup> Re( ( )) Re( ( )) <sup>1</sup> *i i i i cF j c F j* and <sup>1</sup> <sup>1</sup> Im( ( )) Im( ( )) *i i i i cF j c F j* for *i*>*T*, then it can be shown for Re( ( )) *Y j* and Im( ( )) *Y j*that for *n* 0

$$\left| \text{Re}(Y(jo)\_{1\sim T+1}) \right| < \dots < \left| \text{Re}(Y(jo)\_{1\sim T+2n+1}) \right| < \left| \text{Re}(Y(jo)) \right| < \left| \text{Re}(Y(jo)\_{1\sim T+2n}) \right| < \dots < \left| \text{Re}(Y(jo)\_{1\sim T}) \right| $$

$$\left| \text{Im}(Y(jo)\_{1\sim T+1}) \right| < \dots < \left| \text{Im}(Y(jo)\_{1\sim T+2n+1}) \right| < \left| \text{Im}(Y(jo)) \right| < \left| \text{Im}(Y(jo)\_{1\sim T+2n}) \right| < \dots < \left| \text{Im}(Y(jo)\_{1\sim T}) \right| $$

$$\text{Therefore, } \left| Y(jo)\_{1\sim T+1} \right| < \dots < \left| Y(jo)\_{1\sim T+2n+1} \right| < \left| Y(jo) \right| < \left| Y(jo)\_{1\sim T+2n} \right| < \dots < \left| Y(jo)\_{1\sim T} \right|.$$

$$\begin{aligned} \left| Y(joo) \right|^2 &= Y(joo)Y(-joo) \\ &= (F\_0(joo) + cF\_1(joo) + c^2F\_2(joo) + \cdots)(F\_0(-joo) + cF\_1(-joo) + c^2F\_2(-joo) + \cdots) \\ &= \sum\_{n=0,1,2,\dots} c^n \sum\_{i=0}^n F\_i(joo)F\_{n-i}(-joo) \end{aligned}$$

It can be verified that the (2*k*)th terms in the series are positive and the (2*k*+1)th terms are negative for *k*=0,1,2,…. Moreover, it is not difficult to obtain that it needs only the real parts of the terms in *Y j* ( ) to be alternating for <sup>2</sup> *Yj Yj Y j* ( ) ( )( ) to be alternating. (3)

$$\begin{split} \frac{\partial \left| Y(jo) \right|}{\partial c} &= \frac{1}{2} \frac{\partial \left| Y(jo) \right|^{2}}{\left| Y(jo) \right|} \\ &= \frac{1}{2 \left| Y(jo) \right|} \left\{ \text{Re} (F\_{0}(jo) F\_{1}(-jo)) + c \sum\_{n=1,2,\dots} n c^{n-1} \sum\_{i=0}^{n} F\_{i}(jo) F\_{n-i}(-jo) \right\} \\ &= \frac{1}{2 \left| Y(jo) \right|} \left\{ \text{Re} (F\_{0}(jo)) + c \sum\_{n=1,2,\dots} n c^{n-1} \sum\_{i=0}^{n} \left| Y(jo) \right| \right\} \end{split}$$

Since Re( ( ) ( )) 0 1 *Fj F j* <0, there must exist *<sup>c</sup>* <sup>0</sup> such that ( ) <sup>0</sup> *Y j c* for 0 *c c* . This completes the proof. □

### **Appendix D: Proof of theorem 2**

In order to prove Theorem 2, the following lemma is needed, which provides a fundamental technique for the derivation of the main results in Theorem 2 by exploiting the recursive nature of ( ) ,0 (1) ( ( )) ( () ; ) *<sup>n</sup> ns p l lns c* .

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 27

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*c*

x x 1 p

 

*<sup>p</sup> <sup>k</sup> lXi lXi ns c*

1

 

( ( ) 1) ( ( ) ( 1) 1)

 

*<sup>p</sup> <sup>k</sup> lXi lXi p x*

 

*x p i*

*pn p p n*

 can be

1 ,0

*x p i*

*<sup>i</sup> <sup>n</sup>*

*i i*

f {s , ,s }

*px p lX j lX j p x*

( () ; ) *<sup>j</sup> j j*

where, if *<sup>i</sup> x* =0, ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) 1 *<sup>i</sup>*

( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ( ( ) 1) ( ( ) ( 1) 1)

 

*lXi lXi p x p x lXi lXi p x*

*i i*

 

( () ; ) ( )

\*

The recursive terminal of ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) *<sup>i</sup>*

( ) ,0 (1) ( ( )) ( 1) 1 ,0 ,0 ,0 (1) ( ( 1) 1)

( () ; ) ( ( ) ( ) ( ); )

*p n n*

,0

 

*j j Lj j*

*p*

*pn l lp n <sup>i</sup> s c*

*p*

*i*

1

( () ; ) *<sup>i</sup> i i*

*i* 

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*c*

*px p lXi lXi p x*

1 ,0

*<sup>n</sup> <sup>i</sup> x p <sup>i</sup>*

12 p 1 p

*n*

*j j Lj j*

( 1) 1 (1) (( 1) 1) all the different combinations all the different <sup>1</sup> of {x ,x ,...,x } satisfying permutations of

*<sup>p</sup> <sup>x</sup>*

 

*pn l lp n i*

*ns c x p*

( ( ( ) )) ,0

 

*i*

 

all the 2 partitions all the different for satisfying permutations ( ) () o

 

1 1

1 ,0 (1) (( 1) 1)

( ( ),( 1) 1; )

*ns p l lns p n p p p l lp n p l lp n*

*x*

*n*

( )

 

*i*

*j j Lj j*

*x*

*c*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*px p lXi lXi p x*

*i i*

fying

*c cc c*

 

   

*fc p n*

*i i*

*x x*

12 p

*Proof of Lemma 3*.

*n*

 

*<sup>p</sup> <sup>x</sup>*

1

written as

*i* 

1 ,0

*s ss c*

*p*

*fs sc*

  1p j

all the different combinations of {x ,x ,...,x } satis

x x 1, 0 x -1

*x*

*i i*

1 p

*x*

*i*

*k*

(x , ,x )

*c* , otherwise,

1

*i i*

 

<sup>1</sup> <sup>1</sup> <sup>1</sup> ,0 ,0

( ( ) 1) ( ( ) ( ( ( ) ))) ( 1) 1 (1) (( 1) 1) all the p partitions all the different <sup>1</sup> for ( ) permutations of {s , ,s } 1

( ( () )

x x each combination

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

Note that different permutations in each combination have no difference to

*c*

*px p lXi lXi p x*

<sup>i</sup> 1, 0 x n-1

, thus ( 1) 1 ,0 1 ( 1) 1 ( () ; ) *<sup>n</sup>*

*s c*

x x 1 p

,0 ( ( ) 1) ( ( ) ( ( ( ) ))) <sup>1</sup>

*c*

*n*

<sup>1</sup> ( ) ( )

( () ; ) *<sup>i</sup> i i*

<sup>1</sup> ( ) ( )

<sup>2</sup> <sup>0</sup> (1) ( ( )) ,0 ( ( ) 1) ( ( ( ) )) ( ( ) ( ( ( ) ))) <sup>1</sup> ( ( ( ) ); ) *<sup>n</sup>* ( ( ( ) ); *<sup>n</sup>* ) *<sup>p</sup> <sup>i</sup> x p x p <sup>i</sup> <sup>i</sup>*

*s c*

*ax x p l lns ns c x p lXi lXi ns c*

*c* is *<sup>i</sup> x* =1.

*px p lXi lXi p x*

*j*

*<sup>p</sup> <sup>x</sup>*

3,0

*s c*

all the p partitions for ( )

   

; ) *<sup>n</sup>*

*lXi lXi ns c*

 

*px p lXi lXi p x*

**Lemma 3**. Consider a nonlinear parameter denoted by *c*p,q(*k*1,*k*2,…,*k*p+q).

(1) If *p* 2 and *q*=0, then

( ) ,0 (1) ( ( )) ( 1) 1 ,0 (1) (( 1) 1) ( 1) 1 1 1 () 1 ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 (1) (( 1) 1) all the 1 ( () ; ) ( () ; ) ( 1) ( ) ( () ; ) ( ) *i i i n n ns p l lns p n p l l p n p n n l i p i x px p lXi lXi p x pn l lp n i c c H j <sup>c</sup> Lj j* 12 p 1p i 1 p different combinations of {x ,x ,...,x } satisfying x x 1, 0 x n-1 \* 1 p \* ( ( ) 1) ( ( ) ( 1) 1) 1 p all the different 1 permutations of {k , ,k } (x , ,x ) ( ) (k , ,k ) *i i n <sup>p</sup> <sup>x</sup> <sup>k</sup> lXi lXi p x k i <sup>n</sup> j j <sup>n</sup>* 

where,

12 p 1 p ( 1) 1 ,0 (1) (( 1) 1) ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying x x ( () ; ) <sup>1</sup> ( () ; ) ( ) *i i i n p n p l lp n <sup>p</sup> <sup>x</sup> px p lXi lXi p x pn l lp n i c <sup>c</sup> Lj j* i 1 p 1, 0 x n-1 \* 1 p \* ( ( ) 1) ( ( ) ( 1) 1) 1 p all the different 1 permutations of {k , ,k } (x , ,x ) ( ) (k , ,k ) *i i n <sup>p</sup> <sup>x</sup> <sup>k</sup> lXi lXi p x k i <sup>n</sup> j j <sup>n</sup>* 

the termination is 1(1; ) 1 *<sup>i</sup>* . \* 1 1 2 ! (,,) !! ! *k p e <sup>p</sup> nk k nn n* , *n*1+…+*n*e=*p*, *e* is the number of

distinct differentials *k*i appearing in the combination, *n*i is the number of repetitions of *k*i, and a similar definition holds for \* <sup>1</sup> (,,) *x p nx x* .

(2) If *p* 2 , *q*=0 and *k*1=*k*2=…=*k*p=*k*, then

$$\begin{split} & \quad \phi\_{(p-1)n+1} \{ \boldsymbol{c}\_{p,0} (\boldsymbol{\lambda}^\*; o\boldsymbol{\alpha}\_{l(1)} \cdots o\boldsymbol{\alpha}\_{l((p-1)n+1)}) \\ &= \frac{(-1)^{n-1} \prod\_{i=1}^{(p-1)n+1} \{ (jo\_{l(i)})^k H\_1(jo\_{l(i)}) \}}{\prod\_{i=(p-1)n+1} \{ jo\_{l(1)} + \cdots + jo\_{l((p-1)n+1)} \}} \\ & \cdot \sum\_{\substack{\text{all the different combinations} \\ \text{of } \{\overline{\mathbf{x}\_1}, \overline{\mathbf{x}\_2}, \ldots, \overline{\mathbf{x}\_p}\} \text{ satisfying}} \; \alpha\_x^\* \{ \overline{\mathbf{x}\_1}, \ldots, \overline{\mathbf{x}\_p} \} \cdot \prod\_{i=1}^p \phi\_{(p-1)\overline{\mathbf{x}\_i} + 1}^{\sigma} \{ c\_{p,0} (\boldsymbol{\lambda}^\*)^{\overline{\mathbf{x}\_i}}; o\boldsymbol{\alpha}\_{l(\overline{\mathbf{x}\_i} \boldsymbol{\epsilon}) + 1} \cdots o\_{l(\overline{\mathbf{x}\_i} \boldsymbol{\epsilon}) + (p-1)\overline{\mathbf{x}\_i}} \} \\ & \cdot \sum\_{\substack{\text{all the different combinations} \\ \text{of } \{\overline{\mathbf{x}\_1}, \overline{\mathbf{x}\_2}, \ldots, \overline{\mathbf{x}\_p}\} \text{ satisfying}} \; \alpha\_x^\* \{ c\_{p,0} (\boldsymbol{\lambda}^\*)^{\overline{\mathbf{x}\_i}}; o\boldsymbol{\alpha}\_{l(\overline{\mathbf{x}\_i} \boldsymbol{\epsilon}) + 1} \cdots \; \alpha\_x^\* \{ c\_{p,0} (\boldsymbol{\lambda}^\*)^{\overline{\mathbf{x}\_i}}; o\boldsymbol{\alpha}\_{l(\overline{\mathbf{x}\_i} \boldsymbol{\epsilon}) +$$

$$\text{where, if } \overline{\mathfrak{x}}\_i \bullet 0\_\prime \bullet\_{(p-1)\overline{\mathfrak{x}}\_i+1} (\mathfrak{c}\_{p,0}(\cdot)^{\overline{\mathfrak{x}}\_i}; o\_{l(\overline{\mathfrak{X}}(i)+1)} \cdots o\_{l(\overline{\mathfrak{X}}(i)+(p-1)\overline{\mathfrak{x}}\_i+1)}) = 1 \text{, otherwise, } \overline{\mathfrak{x}}\_i$$

$$=\frac{\begin{pmatrix} jo\_{l(p-1)\overline{\pi}\_{i}}(c\_{p,0})^{\overline{\pi}\_{i}};o\_{l(\overline{\chi}(i)+1)}\cdots o\_{l(\overline{\chi}(i)+(p-1)\overline{\pi}\_{i}+1)} \\\hline \\ -\underline{\operatorname{Im}}\_{l(p-1)\overline{\pi}\_{i}+1}(jo\_{l(\overline{\chi}(i)+1)}+\cdots+jo\_{l(\overline{\chi}(i)+(p-1)\overline{\pi}\_{i}+1)} \\\hline \\ \sum\limits\_{\begin{subarray}{c}\text{all the different combinations}\\ \text{of }\chi\_{1},\chi\_{2},\cdots,\chi\_{p}\text{ satisfying} \\\hline \chi\_{1}+\cdots+\chi\_{p}=\overline{\pi}\_{i}-1,0\text{ s.}\ \mathfrak{s}\_{i}\mathfrak{s}\_{i}\end{subarray}}\prod\limits\_{j=1}^{p}\phi^{\sf{r}}\_{(p-1)\underline{\text{r}}\_{i}+1}(c\_{p,0}(\boldsymbol{\chi})^{\overline{\boldsymbol{r}}\_{j}};o\_{l(\overline{\chi}(i)+1)}\cdots o\_{l(\overline{\chi}(i)+(p-1)\underline{\text{r}}\_{j}+1)} \\\hline \\ \sum\limits\_{\begin{subarray}{c}\text{all the different combinations}\\ \begin{subarray}{c}\text{\$\mathfrak{s}\_{1}\$ and \$\mathfrak{s}\_{2}\$ and \$\mathfrak{s}\_{3}\$\text{\$\mathfrak{s}\_{1}\$}\$\\ \end{subarray}}\phi^{\sf{r}}\_{(p-1)\underline{\text{r}}\_{i}+1}(jo\_{p,0}(\boldsymbol{\chi})^{\overline{\mathfrak{s}}\_{i}};o\_{l(\overline{\chi}(i)+1)}\cdots o\_{l(\overline{\chi}(i)+(p-1)\underline{\text{r}}\_{j}+1)} \\\hline \\ \vdots \\\hline \end{pmatrix}$$

The recursive terminal of ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) *<sup>i</sup> i i x px p lXi lXi p x c* is *<sup>i</sup> x* =1.

*Proof of Lemma 3*.

26 Advances on Analysis and Control of Vibrations – Theory and Applications

( ) ,0 (1) ( ( )) ( 1) 1 ,0 (1) (( 1) 1)

( () ; ) ( () ; )

*<sup>c</sup> Lj j*

1 p

1 p all the different 1 permutations of {k , ,k }

( 1) 1 (1) (( 1) 1) all the 1

*k i*

( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying

*k i*

*pn l lp n i*

*pn l lp n i*

**Lemma 3**. Consider a nonlinear parameter denoted by *c*p,q(*k*1,*k*2,…,*k*p+q).

*l i p i x*

> 12 p 1p i

*<sup>n</sup> j j <sup>n</sup>*

12 p 1 p

*<sup>n</sup> j j <sup>n</sup>*

1

<sup>1</sup> (,,) *x p nx x* .

\*

all the different 1

combinations

*n*

x x

*<sup>c</sup> Lj j*

1 p

1 p all the different 1 permutations of {k , ,k }

*n*

 

different combinations of {x ,x ,...,x } satisfying x x 1, 0 x n-1

\* ( ( ) 1) ( ( ) ( 1) 1)

 

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

(x , ,x ) ( ) (k , ,k )

In order to prove Theorem 2, the following lemma is needed, which provides a fundamental technique for the derivation of the main results in Theorem 2 by exploiting the recursive

( () ; ) ( )

<sup>1</sup> ( () ; ) ( )

1 2 ! (,,) !! ! *k p*

i

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

\* ( ( ) 1) ( ( ) ( 1) 1)

(x , ,x ) ( ) (k , ,k )

distinct differentials *k*i appearing in the combination, *n*i is the number of repetitions of *k*i, and

*i*

*n c*

1, 0 x n-1

 

*<sup>p</sup> nk k*

*lXi lXi p x*

*e*

*<sup>p</sup> <sup>x</sup>*

 

*lXi lXi p x*

 

*<sup>p</sup> <sup>x</sup>*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*i i*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*i i*

*px p lXi lXi p x*

*i i*

*i*

*nn n* , *n*1+…+*n*e=*p*, *e* is the number of

1 p ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*i i*

(x , ,x ) ( () ; ) *<sup>i</sup>*

*x px p lXi lXi p x*

*px p lXi lXi p x*

*i i*

*i*

**Appendix D: Proof of theorem 2** 

nature of ( ) ,0 (1) ( ( )) ( () ; ) *<sup>n</sup> ns p l lns*

 *c* .

1 ()

1 p

*c c*

*H j*

 

*n n ns p l lns p n p l l p n*

 

(1) If *p* 2 and *q*=0, then

 

where,

( 1) 1 1

 

*p n n*

1

( 1) 1 ,0 (1) (( 1) 1)

the termination is 1(1; ) 1 *<sup>i</sup>*

a similar definition holds for \*

 

*n p n p l lp n*

*c*

( () ; )

\*

 

 . \*

(2) If *p* 2 , *q*=0 and *k*1=*k*2=…=*k*p=*k*, then

( () ; )

( )

 

*j Hj*

() 1 ()

 

*l i l i*

( 1) 1 ,0 (1) (( 1) 1)

 

*n p n p l lp n*

( 1) 1 (1) (( 1) 1)

12 p 1p i

of {x ,x ,...,x } satisfying x x 1, 0 x n-1 *n*

*pn l lp n*

( 1) [( ) ( )]

( 1) 1 1

*p n n k*

*c*

*i*

1

*Lj j*

1 p

\*

( 1) ( )

 1 ,0 ( ) ,0 (1) ( ( )) ( 1) 1 ,0 ,0 ,0 (1) ( ( 1) 1) 1 ,0 (1) (( 1) 1) all the 2 partitions all the different for satisfying permutations ( ) () o ( () ; ) ( ( ) ( ) ( ); ) ( ( ),( 1) 1; ) *p n ns p l lns p n p p p l lp n p l lp n s ss c c cc c fc p n* 3,0 x x 1 p <sup>1</sup> <sup>1</sup> <sup>1</sup> ,0 ,0 all the p partitions for ( ) f {s , ,s } 1 1 <sup>2</sup> <sup>0</sup> (1) ( ( )) ,0 ( ( ) 1) ( ( ( ) )) ( ( ) ( ( ( ) ))) <sup>1</sup> ( ( ( ) ); ) *<sup>n</sup>* ( ( ( ) ); *<sup>n</sup>* ) *<sup>p</sup> <sup>i</sup> x p x p <sup>i</sup> <sup>i</sup> s c p n n ax x p l lns ns c x p lXi lXi ns c i fs sc s c* 

$$=\frac{1}{L\_{(p-1)n+1}(ja\_{\mathbb{I}(1)}+\cdots+ja\_{\mathbb{I}((p-1)n+1)})}\cdot\sum\_{\substack{\text{all the }p-\text{particles}}}\sum\_{\substack{\text{all the different }p\\\text{permutations}\\ \text{of } (\kappa\_{i\_1},\cdots,\kappa\_{i\_p})}}\left[\prod\_{i=1}^p (ja\_{\mathbb{I}(\overline{\mathbb{J}}(i)+1)}+\cdots+ja\_{\mathbb{I}(\overline{\mathbb{J}}(i)+n(\underline{\mathbb{J}}\_1\{c\_{p,\mathbb{I}}\})^{n-1}))})^{k\_p}\right]$$

$$\cdot\prod\_{i=1}^p \boldsymbol{\upphi}\_{n\left(\mathfrak{s}\_{\overline{\mathbb{J}}\_i\left(\boldsymbol{c}\_{p,\mathbb{I}}\left(\boldsymbol{\upchi}^{-1}\right)\right)\right)}}\left\{\boldsymbol{c}\_{\mathfrak{T}\_i\left(\boldsymbol{c}\_{p,\mathbb{I}}\left(\boldsymbol{\upchi}^{-1}\right)\right)},\boldsymbol{c}\_{\mathfrak{U}\left(\overline{\mathbb{J}}(i)+1\right)}\cdots,\boldsymbol{c}\_{\mathfrak{U}\left(\overline{\mathbb{J}}(i)+n(\underline{\mathbb{J}}\_1\{c\_{p,\mathbb{I}}\})^{n-1}\right)}\right\}$$

$$=\frac{1}{L\_{\{p-1\}\times 1}(j\alpha\_{l(1)}+\cdots+j\alpha\_{l((p-1)\times 1)})}\cdot\sum\_{\substack{\text{all the different combinations}\\ \text{of } \{\overline{\mathfrak{u}}\_1, \overline{\mathfrak{u}}\_2, \dots, \overline{\mathfrak{u}}\_p\} \text{ satisfying}}}\cdot\sum\_{\substack{\text{all the different}\\ \text{permutations of}\\ \text{is}}}\left[\prod\_{l=1}^p (j\alpha\_{l(\overline{\mathfrak{u}}\_l)+1}+\cdots+j\alpha\_{l(\overline{\mathfrak{u}}\_l)\times(p-1)\overline{\mathfrak{u}}\_l+1})^{\frac{1}{p}}\right]$$

$$\cdot\prod\_{l=1}^p \phi\_{(p-1)\overline{\mathfrak{u}}\_1+1}(\mathbf{c}\_{p,0}) \cdot \mathbf{1}\_{l,0}^{\mathfrak{u}\_1} \cdot \mathbf{o}\_{l(\overline{\mathfrak{u}}\_l)+1} \cdots \mathbf{o}\_{l(\overline{\mathfrak{u}}\_l)+(p-1)\overline{\mathfrak{u}}\_l+1} \cdot 1\_{l}$$

Note that different permutations in each combination have no difference to ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) 1 ( () ; ) *<sup>i</sup> i i <sup>p</sup> <sup>x</sup> px p lXi lXi p x i c* , thus ( 1) 1 ,0 1 ( 1) 1 ( () ; ) *<sup>n</sup> pn p p n c* can be written as

12 p 13 i ( 1) 1 ,0 1 ( 1) 1 ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying x x 1, 0 x n- ( () ; ) <sup>1</sup> ( () ; ) ( ) *i i i n pn p p n <sup>p</sup> <sup>x</sup> px p lXi lXi p x pn l lp n i n c <sup>c</sup> Lj j* 1 ( ( ) 1) ( ( ) ( 1) 1) all the different 1 permutations of each combination ( ) *<sup>i</sup> i <sup>p</sup> <sup>k</sup> lXi lXi p x i j j* 12 p 1p i ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying x x 1, 0 x n-1 \* 1 p \* 1 p <sup>1</sup> ( () ; ) ( ) (x , ,x ) ( (k , ,k ) *i i i <sup>p</sup> <sup>x</sup> px p lXi lXi p x pn l lp n i n x l k <sup>c</sup> Lj j <sup>n</sup> <sup>j</sup> <sup>n</sup>* 1 p ( ( ) 1) ( ( ) ( 1) 1) all the different 1 permutations of {k , ,k } ) *i i <sup>p</sup> <sup>k</sup> X i lXi p x i j* 

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 29

*lXi lXi p x*

 

*i*

(A3)

(A4)

(A5)

*i*

*i*

*i*

appear individually in the same form in the subsequent recursion. At the end of the recursion, all the frequency variables should have appeared in this form. Thus these terms

\* ( ( ) 1) ( ( ) ( 1) 1)

*x lXi lXi p x*

(x , ,x ) ( ) (k , ,k )

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

1 p ( ( ) 1) ( ( ) ( 1) 1)

*<sup>p</sup> <sup>k</sup>*

 

( ) 1 ( ) ( 1) 1 ,0 (1) (( 1) 1)

*<sup>p</sup> <sup>x</sup>*

 

*i*

1 p ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*i i*

(x , ,x ) ( () ; ) *<sup>i</sup>*

*x px p lXi lXi p x*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*px p lXi lXi p x*

*i i*

*li li p n p l l p n*

*j j*

(x , ,x ) ( )

( 1) [( ) ( )] ( ( ) ; )

<sup>1</sup> ( () ; ) ( )

*<sup>p</sup> k x*

*i*

*c* , otherwise,

*n c*

 

<sup>i</sup> , 0 x n-1 \* (1 ( )) 1 p ( ( ) 1) ( ( ) ( 1) 1)

(x , ,x ) ( ) *i i*

*i*

*<sup>p</sup> <sup>x</sup>*

 

can also be brought out as common factors if *k*1=*k*2=…=*k*p. In the case of *k*1=*k*2=…=*k*p=*k*,

1 p

1

*n jj*

*n k n*

 

12 p 1 p

*n*

x x1

*x lXi lXi p x*

\*

*i i*

*px p lXi lXi p x*

all the different combinations 1

where, if *<sup>i</sup> x* =0, ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) 1 *<sup>i</sup>*

*x*

ing

*j Hj c*

*i*

1 p all the different 1 permutations of {k , ,k }

*k i*

Therefore (A1) and (A2) can be written, if *k*1=*k*2=…=*k*p, as

( 1) 1 ,0 (1) (( 1) 1)

 

( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying

1

*i n jj*

 

*pn l lp n i*

*n p n p l lp n*

( () ; )

*<sup>c</sup> Lj j*

( 1) 1

*p n*

*c*

( 1) 1 ,0 (1) (( 1) 1)

 

(A4) can be further written as

( 1) 1 ,0 (1) (( 1) 1)

 

*n p n p l lp n*

*Lj j*

*c*

( () ; ) 1 ( )

 

( 1) 1 (1) (( 1) 1)

*pn l lp n*

12 p

of {x ,x ,...,x } satisfy

1p i

x x 1, 0 x n-1 *n*

*n p n p l lp n*

*c*

( () ; )

  *i*

1

\*

*n*

*n*

1 p

\*

\* <sup>1</sup> (,,) *x p nx x* and \* <sup>1</sup> (,,) *k p nk k* are the numbers of the corresponding combinations involved, which can be obtained from the combination theory and can also be referred to [14]. Inspection of the recursion in the equation above, it can be seen that there are (*p*-1)n +1 <sup>1</sup>( ) *H j<sup>i</sup>* with different frequency variable at the end of the recursion. Thus they can be brought out as a common factor. This gives

$$\left(\boldsymbol{\varrho}\_{\left(p-1\right)n+1}\left(\boldsymbol{c}\_{p,0}\left(\cdot\right)^{\boldsymbol{n}};\boldsymbol{o}\_{\left(1\right)}\cdots\boldsymbol{o}\_{\left(\left(p-1\right)n+1\right)}\right) = \left(-1\right)^{\boldsymbol{n}}\prod\_{i=1}^{\left(p-1\right)n+1}H\_{1}\left(\boldsymbol{i}\boldsymbol{o}\_{\left(i\right)}\right)\cdot\boldsymbol{o}\_{\left(p-1\right)n+1}^{\boldsymbol{\ell}}\left(\boldsymbol{c}\_{p,0}\left(\cdot\right)^{\boldsymbol{n}};\boldsymbol{o}\_{\left(1\right)}\cdots\boldsymbol{o}\_{\left(\left(p-1\right)n+1\right)}\right) \tag{A1}$$

where,

$$\begin{split} \phi\_{(p-1)n+1}^{\*}\big(c\_{p,0}(\boldsymbol{\upbeta}^{n});\alpha\_{\mathbb{I}(1)}\cdots\alpha\_{\mathbb{I}((p-1)n+1)}\big) \\ =\frac{1}{L\_{(p-1)n+1}(ja\_{\mathbb{I}(1)}+\cdots+ja\_{\mathbb{I}((p-1)n+1)})} \cdot \sum\_{\substack{\text{all the different combinations } i=1 \\ \text{if } \overline{\mathfrak{T}}\_{1},\overline{\mathfrak{T}}\_{2},\cdots,\overline{\mathfrak{T}}\_{n}\text{ satisfying}} \prod\_{i=1}^{p} \phi\_{[p-1]\overline{\mathfrak{T}}\_{i}+1}^{\circ}\big(c\_{p,0}(\boldsymbol{\upbeta}^{\mathsf{T}});\alpha\_{\mathbb{I}(\overline{\mathfrak{T}}\_{i})\mapsto 1}\cdots\alpha\_{\mathbb{I}(\overline{\mathfrak{T}}\_{i})\mapsto (p-1)\overline{\mathfrak{T}}\_{i}+1}\big) \\ \cdot \frac{n\_{\times}^{\*}\left(\overline{\mathfrak{T}}\_{1},\cdots,\overline{\mathfrak{T}}\_{p}\right)}{n\_{k}^{\*}\left(\mathbf{k}\_{1},\cdots,\mathbf{k}\_{p}\right)} \cdot \sum\_{\substack{\text{all the different } i=1 \\ \text{all the different } i=1}} \prod\_{i=1}^{p} \Big(j\alpha\_{l\_{i}[\overline{\mathfrak{T}}\_{i}]+1} + \dots + j\alpha\_{l\_{i}[\overline{\mathfrak{T}}\_{i}]+(p-1)\overline{\mathfrak{T}}\_{i}+1}\big)^{k\_{i}} \end{split} \tag{A2}$$

the termination is 1(1; ) 1 *<sup>i</sup>* . Note that when *<sup>i</sup> <sup>x</sup>* =0, there is a term ( ( ) 1) ( ) *<sup>i</sup> <sup>k</sup> lXi j*appearing

$$\text{from } \frac{n\_{\mathbf{x}}^{\star}(\overline{\mathbf{x}}\_{1}, \cdots, \overline{\mathbf{x}}\_{p})}{n\_{\mathbf{k}}^{\star}(\mathbf{k}\_{1}, \cdots, \mathbf{k}\_{p})} \cdot \sum\_{\substack{\text{all the different} \\ \text{permutations of} \\ (\mathbf{k}\_{1}, \cdots, \mathbf{k}\_{p})}} \prod\_{l=1}^{p} (ja\_{l(\overline{\mathbf{x}}(i) + 1)} + \cdots + ja\_{l(\overline{\mathbf{x}}(i) + (p-1)\overline{\mathbf{x}}\_{i} + 1)})^{k\_{i}}. \text{ It can be verified that}$$

in each recursion of ( 1) 1 ,0 (1) (( 1) 1) ( () ; ) *<sup>n</sup> p n p l lp n c* , there may be some frequency variables appearing individually in the form of ( ( ) 1) ( ) *<sup>i</sup> <sup>k</sup> lXi j*, and these variables will not appear individually in the same form in the subsequent recursion. At the end of the recursion, all the frequency variables should have appeared in this form. Thus these terms can also be brought out as common factors if *k*1=*k*2=…=*k*p. In the case of *k*1=*k*2=…=*k*p=*k*,

$$\frac{n\_{\mathbf{x}}^{\*}\left(\overline{\mathbf{x}}\_{1},\cdots,\overline{\mathbf{x}}\_{\mathbf{p}}\right)}{n\_{\mathbf{k}}^{\*}\left(\mathbf{k}\_{1},\cdots,\mathbf{k}\_{\mathbf{p}}\right)}\cdot\sum\_{\substack{\text{all the different}\\ \text{permutations of}\\ \{\overline{\mathbf{k}}\_{1},\cdots,\mathbf{k}\_{\mathbf{p}}\}}}\prod\_{i=1}^{p}\left(j o\_{l(\overline{\mathbf{X}}(i)+1)}+\cdots+j o\_{l(\overline{\mathbf{X}}(i)+\{p-1\}\overline{\mathbf{x}}\_{i}+1)}\right)^{k\_{i}}$$

$$= n\_{\mathbf{x}}^{\*}\left(\overline{\mathbf{x}}\_{1},\cdots,\overline{\mathbf{x}}\_{\mathbf{p}}\right)\cdot\prod\_{i=1}^{p}\left(j o\_{l(\overline{\mathbf{X}}(i)+1)}+\cdots+j o\_{l(\overline{\mathbf{X}}(i)+\{p-1\}\overline{\mathbf{x}}\_{i}+1)}\right)^{k\_{i}}$$

Therefore (A1) and (A2) can be written, if *k*1=*k*2=…=*k*p, as

$$\begin{split} & \left( \boldsymbol{\varrho}\_{(p-1)n+1} \{ \boldsymbol{c}\_{p,0} \} \right) \boldsymbol{\iota} ; \boldsymbol{o}\_{l\{1\}} \cdots \boldsymbol{o}\_{l\{(p-1)n+1\}} \boldsymbol{\iota} \\ &= \{ -1 \} ^{n} \prod\_{i=1}^{(p-1)n+1} \{ (j\boldsymbol{o}\_{l\{i\}}) ^{k} H\_{1} (j\boldsymbol{o}\_{l\{i\}}) \} \cdot \boldsymbol{\varrho}'\_{(p-1)n+1} \{ \boldsymbol{c}\_{p,0} \} \boldsymbol{\iota} ; \boldsymbol{o}\_{l\{1\}} \cdots \boldsymbol{o}\_{l\{(p-1)n+1\}} \} \end{split} \tag{A3}$$

12 p 1 p ( 1) 1 ,0 (1) (( 1) 1) ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying x x1 ( () ; ) <sup>1</sup> ( () ; ) ( ) *i i i n p n p l lp n <sup>p</sup> <sup>x</sup> px p lXi lXi p x pn l lp n i n c <sup>c</sup> Lj j* <sup>i</sup> , 0 x n-1 \* (1 ( )) 1 p ( ( ) 1) ( ( ) ( 1) 1) 1 (x , ,x ) ( ) *i i i <sup>p</sup> k x x lXi lXi p x i n jj* (A4)

(A4) can be further written as

28 Advances on Analysis and Control of Vibrations – Theory and Applications

( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying

( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying

*pn l lp n i*

(x , ,x ) ( (k , ,k )

1 p

*<sup>n</sup> <sup>j</sup> <sup>n</sup>*

*c H j*

( 1) 1 (1) (( 1) 1) all the different combinations <sup>1</sup> of {x ,x ,...,x } satisfying

*<sup>c</sup> Lj j*

1 p

1 p all the different 1 permutations of {k , ,k }

*k i*

*pn l lp n i*

*<sup>c</sup> Lj j*

all the different 1 permutations of each combination

*<sup>c</sup> Lj j*

all the different 1 permutations of {k , ,k }

> 12 p 1 p

*n*

x x1

*<sup>n</sup> j j <sup>n</sup>*

\* ( ( ) 1) ( ( ) ( 1) 1)

 

(x , ,x ) ( ) (k , ,k )

*p n p l lp n*

 

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

*pn l lp n i*

( 1) 1 ,0 1 ( 1) 1

*n pn p p n*

<sup>1</sup> (,,) *x p nx x* and \*

*c*

\*

<sup>1</sup>( ) *H j*

where,

from

( () ; )

\*

\*

*x*

*k*

 

( 1) 1 ,0 (1) (( 1) 1)

 

the termination is 1(1; ) 1 *<sup>i</sup>* 

1 p

*n p n p l lp n*

*c*

\*

*n*

*n*

( () ; )

\*

 

1 p

1 p

in each recursion of ( 1) 1 ,0 (1) (( 1) 1) ( () ; ) *<sup>n</sup>*

variables appearing individually in the form of ( ( ) 1) ( ) *<sup>i</sup> <sup>k</sup>*

1 p all the different 1 permutations of {k , ,k }

*k i*

1 p

 

brought out as a common factor. This gives

 

1 p

 

 

> 12 p 13 i

x x 1, 0 x n-

*n*

*i*

12 p 1p i

x x 1, 0 x n-1

*i*

 

( 1) 1 ( 1) 1 ,0 (1) (( 1) 1) 1 ( ) ( 1) 1 ,0 (1) (( 1) 1) 1

 

*p n nn n p n p l lp n li p n p l l p n i*

*n*

 

<sup>1</sup> ( () ; ) ( )

1

*j j*

<sup>1</sup> ( () ; ) ( )

*j*

<sup>1</sup> (,,) *k p nk k* are the numbers of the corresponding combinations involved,

*<sup>p</sup> <sup>x</sup>*

*lXi lXi p x*

 

(A1)

*l*

which can be obtained from the combination theory and can also be referred to [14]. Inspection of the recursion in the equation above, it can be seen that there are (*p*-1)n +1

*<sup>i</sup>* with different frequency variable at the end of the recursion. Thus they can be

( () ; ) ( 1) ( ) ( ( ) ; )

<sup>1</sup> ( () ; ) ( )

(x , ,x ) ( ) (k , ,k )

*j j*

i

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

\* ( ( ) 1) ( ( ) ( 1) 1)

, 0 x n-1

. Note that when *<sup>i</sup> <sup>x</sup>* =0, there is a term ( ( ) 1) ( ) *<sup>i</sup> <sup>k</sup>*

*lXi lXi p x*

 

> *lXi j*

. It can be verified that

( ( ) 1) ( ( ) ( 1) 1)

( ) *<sup>i</sup>*

 

*<sup>p</sup> <sup>k</sup> lXi lXi p x*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

) *i i*

 *c*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*px p lXi lXi p x*

*i i*

*i i*

*i*

*i*

*c* , there may be some frequency

*lXi j*

, and these variables will not

appearing

*i*

*px p lXi lXi p x*

*i i*

(A2)

*px p lXi lXi p x*

*i i*

*i*

*i*

*i*

*<sup>p</sup> <sup>x</sup>*

*<sup>p</sup> <sup>x</sup>*

( ( ) 1) ( ( ) ( 1) 1)

*<sup>p</sup> <sup>k</sup> X i lXi p x*

$$=\frac{\rho\_{(p-1)n+1}^{\*}(c\_{p,0}(\boldsymbol{\upbeta})^{n};o\_{l(1)}\cdots o\_{l((p-1)n+1)})}{} \tag{A5}$$

$$=\frac{-1}{L\_{(p-1)n+1}(j\alpha\_{l(1)}+\cdots+j\alpha\_{l((p-1)n+1)})} \tag{A6}$$

$$\begin{array}{c} \sum \quad n\_{x}^{\*}\left(\overline{\mathbf{x}}\_{1},\cdots,\overline{\mathbf{x}}\_{p}\right)\cdot\prod\boldsymbol{\upbeta}^{p}\_{(p-1)\overline{\mathbf{x}}\_{i}+1}\left(c\_{p,0}(\boldsymbol{\upbeta})^{\overline{\mathbf{x}}\_{i}};o\_{l(\overline{\mathbf{x}}\_{i})+1}\cdots o\_{l(\overline{\mathbf{x}}\_{i})+(p-1)\overline{\mathbf{x}}\_{i}+1}\right) \\ \quad \times[\overline{\mathbf{x}}\_{1},\cdots,\overline{\mathbf{x}}\_{p}]\cdot\prod\boldsymbol{\upgamma}\_{(p-1)\overline{\mathbf{x}}\_{i}+1}\left(c\_{p,0}(\boldsymbol{\upgamma})^{\overline{\mathbf{x}}\_{i}};o\_{l(\overline{\mathbf{x}}\_{i})+1}\cdots o\_{l(\overline{\mathbf{x}}\_{i})+(p-1)\overline{\mathbf{x}}\_{i}+1}\right) \\ \quad \times[\overline{\mathbf{x}}\_{1},\cdots,\overline{\mathbf{x}}\_{p}]\cdot\prod\boldsymbol{\upgamma}\_{(p-1)\overline{\mathbf{x}}\_{i}+1}\left(c\_{p,0}(\boldsymbol{\upgamma})^{\overline{\mathbf{x}}\_{i}};o\_{l(\overline{\mathbf{x}}\_{i})+1}\cdots o\_{l(\overline{\mathbf{x}}\_{i})+1}\cdots o\_{l(\overline{\mathbf{x}}\_{i})+1}\right) \\ \quad \times[\overline{\mathbf{x}}\_{1},\cdots,\overline{\mathbf{x}}\_{p}]\cdot\prod\boldsymbol{\upgamma}\_{$$

where, if *<sup>i</sup> x* =0, ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( () ; ) 1 *<sup>i</sup> i i x px p lXi lXi p x c* , otherwise,

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ( () ( () ; ) ( ) ( () ; ) ( ) ( *i i i i i i i i i x px p lXi lXi p x k x lXi lXi p x px p lXi lXi p x k lXi lXi p x p x lXi c jj c j j L j* 12 p 1p i \* 1 p 1) ( ( ) ( 1) 1) all the different combinations of {x ,x ,...,x } satisfying x x 1, 0 x -1 (1 ( )) ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ,0 ( ( ) 1) (x , ,x ) ) ( ) ( () ; *i i i i i i i i x lXi p x x x k x x lX i lX i p x px p lX i n j j j c* 12 p 1p i ( ( ) ( 1) 1) 1 ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ( ( ) 1) ( ( ) ( 1) 1) \* 1 p all the different combinations of {x ,x ,...,x } satisfying x x 1, 0 x - ) ( ) ( ) (x , ,x ) *i i i i i i p lX i p x i k lXi lXi p x p x lXi lXi p x x x x j j Lj j n* ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) 1 1 ( () ; ) *<sup>i</sup> i i <sup>p</sup> <sup>x</sup> px p lX i lX i p x i c* 

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 31

*<sup>c</sup>* by using the result in

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*i i*

*lXi lXi p x*

 

*i*

*const*

*i*

*i i*

*i*

*j j*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*px p lXi lXi p x*

*i i*

*lXi lXi p x*

 

*i*

( () ; )

*n pn p k k*

 

*<sup>p</sup> <sup>x</sup>*

*c*

\* ( ( ) 1) ( ( ) ( 1) 1)

(x , ,x ) ( ) (k , ,k )

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

In the equation above, replacing 1 ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n*

( 1) 1

*p n*

*i*

 

*Lj j*

1

all the different combinations 1

..,x } satisfying x x 1, 0 x n-1 \*

1 p

p 1p i

*n*

1 ()

( )

sgn ( () ; )

*<sup>p</sup> <sup>x</sup>*

*c*

\* ( ( ) 1) ( ( ) ( 1) 1)

has no effect on the equality above according to Lemma 2, then

(x , ,x ) ( ) (k , ,k )

*<sup>p</sup> <sup>x</sup> <sup>k</sup>*

*c px p lXi lXi p x i*

1 p

*i*

1 p

1 p all the different 1 permutations of {k , ,k }

*k i*

*<sup>n</sup> j j <sup>n</sup>*

(2). If additionally, *k*1=*k*2=…=*k*p=*k* in *c*p,0(.), then using the result in Lemma 3, (22) can be

1 p all the different 1 permutations of {k , ,k }

*k i*

( )

*l i*

 

( 1) 1 (1) (( 1) 1)

 

*H j*

*pn l lp n*

1 ( 1) 1

*k k p n*

1 ( 1) 1

( 1) 2

*p n*

*i*

1

*k k p n*

 

sgn

Note that

written as

( 1) 2

*p n*

() ()

( )

1 1 1 ( 1) 1

*Hj Hj*

*i p n*

*L j*

 

*<sup>c</sup> <sup>n</sup>*

 

1 2

*n*

*n*

12 p 1p i

 

\*

2

1

the equation above is equivalent to (22).

*H j*

( )

all the different combinations 1 of {x ,x ,...,x } satisfying x x 1, 0 x n-1

1 p

2

of {x ,x ,.

Lemma 3 and noting (*p*-1)*n*+1 is an odd integer, it can be obtained that

The recursive terminal of (A6) is *<sup>i</sup> x* =1. Substituting (A2) into (A1) gives the first point of the lemma and substituting (A5) and (A6) into (A3) yields the first point of the lemma. This completes the proof. □

Now proceed with the proof of Theorem 2. For convenience, denote

$$\operatorname{sgn}\_c(\nu\_1)^\* \operatorname{sgn}\_c(\nu\_2) = \operatorname{sgn}\_c(\nu\_1 \nu\_2) = \left\lfloor \operatorname{sgn}\_r(\operatorname{Re}(\nu\_1 \nu\_2)) \right\rfloor \quad \operatorname{sgn}\_r(\operatorname{Im}(\nu\_1 \nu\_2)) \left\lfloor \right\rfloor$$

for any 1 2 ,ℂ.

*Proof of Theorem 2*. (1). From Lemma 1, any asymmetric <sup>1</sup> ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n n pn p k k <sup>c</sup>* is sufficient for the computation of ( 1) 1( ) *p n F j* . It can be obtained that

$$\operatorname{sgn}\_c(\tilde{F}\_{(p-1)n+1}(j\Omega)) = \operatorname{sgn}\_c(-j(\frac{F\_d}{2})^{(p-1)n+1}) \ast \operatorname{sgn}\_c(\sum\_{a\_{k\_1} + \dots + a\_{k\_{(p-1)n+1}} = \Omega} \phi\_{(p-1)n+1}(c\_{p,0}(\cdot)^n; a\_{k\_1}, \dots, a\_{k\_{(p-1)n+1}}))$$

From Lemma 2, ( 1) 1 sgn ( ( ) ) <sup>2</sup> *d p n c F j* has no effect on the alternating nature of the sequence ( 1) 1( ) *p n F j* for *n*=1,2,3,…. Hence, (21a-c) is an alternating series with respect to *c*p,0(.) if and only if the sequence 1 ( 1) 1 1 ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n k k p n n pn p k k c* for *n*=1,2,3,… is

alternating. This is equivalent to

$$\operatorname{sgn}\_{\varepsilon} \left( \sum\_{a\_{k\_1} + \dots + a\_{k\_{\{p-1\} n+1}} = \Omega} (-1)^{n-1} \, \rho\_{\{p-1\} n+1} \{c\_{p,0} \{\cdot\}^n; a\_{l(1)} \cdots a\_{l(\{p-1\} n+1} \} \right) = \operatorname{const}$$

In the equation above, replacing 1 ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n n pn p k k <sup>c</sup>* by using the result in Lemma 3 and noting (*p*-1)*n*+1 is an odd integer, it can be obtained that

30 Advances on Analysis and Control of Vibrations – Theory and Applications

( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*k x lXi lXi p x px p lXi lXi p x k*

1) ( ( ) ( 1) 1) all the different combinations

*lX i lX i p x px p lX i*

1

*i* 

 

 

*i i i i*

12 p 1p i

> 

(1 ( )) ( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ,0 ( ( ) 1)

 

*i i*

*<sup>p</sup> <sup>x</sup>*

The recursive terminal of (A6) is *<sup>i</sup> x* =1. Substituting (A2) into (A1) gives the first point of the lemma and substituting (A5) and (A6) into (A3) yields the first point of the lemma. This

*Proof of Theorem 2*. (1). From Lemma 1, any asymmetric <sup>1</sup> ( 1) 1 ( 1) 1 ,0 ( () ; , , ) *p n*

( 1) 1 ( 1) 1 ,0 sgn ( ( )) sgn ( ( ) ) sgn ( ( ( ) ; , , )) <sup>2</sup> *p n*

for *n*=1,2,3,…. Hence, (21a-c) is an alternating series with respect to *c*p,0(.) if and

*n n <sup>c</sup> p n p l lp n c const*

 

*c pn c c pn p k k*

of {x ,x ,...,x } satisfying x x 1, 0 x -1

*i i i*

*k x x*

*i i*

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

 1 2 1 2 sgn (Re( )) sgn (Im( )) *r r* 

. It can be obtained that

1 ( 1) 1

*j* has no effect on the alternating nature of the sequence

( 1) 1 ,0 ( () ; , , ) *p n*

 

*n pn p k k c*

> 

*k k p n d p n n*

 

*x x*

(x , ,x ) )

\*

 

*i i*

*c*

*px p lX i lX i p x*

( () ; ) *<sup>i</sup>*

 

*n pn p k k*

 *<sup>c</sup>* is

> 

for *n*=1,2,3,… is

1 ( 1) 1

*x*

*n*

1 p

( ( ) ( 1) 1)

*lX i p x*

)

*i*

( ) ( () ; )

( ) ( () ;

*k*

*j j c*

1 p

Now proceed with the proof of Theorem 2. For convenience, denote

( 1) 1

only if the sequence 1 ( 1) 1 1 ( 1) 1

1 ( 1) 1 ,0 (1) (( 1) 1) sgn ( 1) ( ( ) ; )

*k k p n*

 

 

*Fj j c*

<sup>1</sup> <sup>2</sup> 1 2 sgn ( ) \* sgn ( ) sgn ( ) *cc c*

*F*

*d p n*

*F*

1 ( 1) 1

 

*k k p n*

  *i*

(x , ,x )

*i*

*i*

*lXi p x*

 

( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1)

*px p lXi lXi p x*

*i i*

*i*

 

*x*

*j j*

( 1) 1 ( ()

12 p 1p i

completes the proof. □

for any 1 2 ,ℂ.

( 1) 1( ) *p n F j*

all the different combinations of {x ,x ,...,x } satisfying x x 1, 0 x -

*j j Lj j*

*i i*

sufficient for the computation of ( 1) 1( ) *p n F j*

From Lemma 2, ( 1) 1 sgn ( ( ) ) <sup>2</sup>

alternating. This is equivalent to

*c*

*x x*

*p x lXi*

*i*

*p*

1

*i*

*L j*

*c*

(

( ( ) 1) ( ( ) ( 1) 1)

( ( ) 1) ( ( ) ( 1) 1) ( 1) 1 ( ( ) 1) ( ( ) ( 1) 1)

1

 

*i i*

 

( )

*lXi lXi p x p x lXi lXi p x*

*j*

\*

*x*

*n*

( )

 

 

*jj c*

( )

*lXi lXi p x*

( () ; )

1 ( 1) 1 1 2 ( 1) 1 1 () 1 ( 1) 1 (1) (( 1) 1) ( 1) 1 ,0 ( ( ) 1) ( ( ) ( 1) 1) all the different combinations 1 of {x ,x ,. ( ) ( ) sgn ( () ; ) *k k p n i i i p n l i i pn l lp n <sup>p</sup> <sup>x</sup> c px p lXi lXi p x i H j Lj j c* p 1p i 1 p ..,x } satisfying x x 1, 0 x n-1 \* 1 p \* ( ( ) 1) ( ( ) ( 1) 1) 1 p all the different 1 permutations of {k , ,k } (x , ,x ) ( ) (k , ,k ) *i i n <sup>p</sup> <sup>x</sup> <sup>k</sup> lXi lXi p x k i n j j n* 

$$=\text{sgn}\_{\boldsymbol{\varepsilon}}\left[\begin{aligned} &\left(\begin{aligned} &H\_{1}(\{\boldsymbol{\Omega}\} & \prod\_{i=1}^{(p-1)n} \Big[\boldsymbol{H}\_{1}(\{\boldsymbol{\Omega}\} \Big] \Big] & \text{\$ \scriptstyle \! 0 \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!$$

Note that ( 1) 2 2 1 1 ( ) *p n i H j* has no effect on the equality above according to Lemma 2, then the equation above is equivalent to (22).

(2). If additionally, *k*1=*k*2=…=*k*p=*k* in *c*p,0(.), then using the result in Lemma 3, (22) can be written as

$$\operatorname\*{sgn}\_{c}\left[\begin{array}{c} \left(j\Omega\right)^{k}H\_{1}\left(j\Omega\right) \\ L\_{(p-1)n+1}\left(j\Omega\right)\_{\operatorname{od}\_{1},\cdots,\operatorname{od}\_{l\_{(p-1)n+1}}}\sum\_{\begin{subarray}{c}\mathfrak{a}\text{all the different combinations} \\ \mathfrak{a}\in\mathbb{F}\_{1},\mathfrak{T}\_{2},\ldots,\mathfrak{T}\_{p}\end{subarray}}\sum\_{\begin{subarray}{c}\mathfrak{a}\in\operatorname{al}\_{\operatorname{v}}\left(\mathfrak{T}\_{1},\cdots,\mathfrak{T}\_{p}\right) \\ \mathfrak{T}\_{1}+\cdots+\mathfrak{T}\_{p}=n-1,\ 0\le\mathfrak{T}\_{i}\le\mathfrak{n}-1 \end{subarray}}\left[n\_{\mathfrak{a}}\left(\overline{\mathfrak{T}}\_{1},\cdots,\overline{\mathfrak{T}}\_{p}\right)}^{\ast}\right]\right] = \operatorname\*{const}$$

Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain 33

[7] Jing X.J., Lang Z.Q., and Billings S. A., "Mapping from parametric characteristics to generalised frequency response functions of nonlinear systems", International Journal

[8] Jing X.J., Lang Z.Q., Billings S. A. and Tomlinson G. R., Frequency domain analysis for suppression of output vibration from periodic disturbance using nonlinearities. Journal

[9] Lang Z.Q., and Billings S. A. "Output frequency characteristics of nonlinear systems".

[10] Leonov G.A., Ponomarenko D.V. and Smirnova V.B. Frequency-domain methods for nonlinear analysis, theory and applications. World Scientific Publishing Co Pte Ltd,

[11] Ljung, L. System Identification: Theory for the User (second edition). Prentice Hall,

[12] Pavlov A., van de Wouw N., and Nijmeijer H., Frequency Response Functions for Nonlinear Convergent Systems, IEEE Trans. Automatic Control, Vol 52, No 6, 1159-

[13] Nuij P.W.J.M., Bosgra O.H., Steinbuch M. "Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses". Mechanical

[14] Peyton-Jones J.C. "Simplified computation of the Volterra frequency response functions of nonlinear systems". Mechanical systems and signal processing, Vol 21, Issue 3, pp

[15] Pintelon R. and Schoukens J., System Identification: A Frequency Domain Approach,

[16] Rugh W.J., Nonlinear System Theory: the Volterra/Wiener Approach, Baltimore,

[17] Sandberg I. W., The mathematical foundations of associated expansions for mildly

[18] Solomou, M. Evans, C. Rees, D. Chiras, N. "Frequency domain analysis of nonlinear systems driven by multiharmonic signals", Proceedings of the 19th IEEE conference on

[19] Schetzen M., The Volterra and Wiener Theory of Nonlinear Systems, J. Wiley and Sons,

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[21] Orlowski P., Frequency domain analysis of uncertain time-varying discretetime systems, Circuits Systems Signal Processing, Vol. 26, No. 3, 2007, PP. 293–310,

[22] Jing X.J., Lang Z.Q., and Billings S. A., Parametric Characteristic Analysis for Generalized Frequency Response Functions of Nonlinear Systems. Circuits Syst Signal

nonlinear systems, IEEE Trans. Circuits Syst., CAS-30, pp441-455, 1983

Instrumentation and Measurement Technology, Vol 1, pp: 799- 804, 2002

of Control, 81(7), 1071-1088, 2008a

Upper Saddle River. 1987 and 1999

Singapore, 1996

1165, 2007

1980

2007

1452-1468, April 2007

Dover, New York, 1959

IEEE Press, Piscataway, NJ, 2001

of Sound and Vibration, 314, 536-557, 2008b

International Journal of Control, Vol. 64, 1049-1067, 1996

Systems and Signal Processing Vol 20, pp1883–1904, 2006

Maryland, U.S.A.: Johns Hopkins University Press, 1981

Process, DOI: 10.1007/s00034-009-9106-7, 2009

From Lemma 2, ( )*<sup>k</sup> j* has no effect on this equation. Then the equation above is equivalent to

$$\operatorname\*{sgn}\_{c} \left[ \begin{aligned} & \left( \frac{H\_{1}(j\Omega)}{L\_{(p-1)n+1}(j\Omega)} \sum\_{\substack{a\_{\overline{\mathbf{a}}\_{1}}, \dots, \overline{\mathbf{a}}\_{\overline{\mathbf{a}}\_{(p-1)n+1}} \in \Omega \text{ all the different combinations}}} \sum\_{\substack{\mathbf{a}\_{\mathbf{x}} \text{ of } \{\overline{\mathbf{x}}\_{1}, \dots, \overline{\mathbf{x}}\_{p}\} \text{ satisfying}}} \left[ n\_{\mathbf{x}}^{\ast} \{\overline{\mathbf{x}}\_{1}, \dots, \overline{\mathbf{x}}\_{p} \} \right] \\ & \qquad \qquad \qquad \overline{\mathbf{x}}\_{1} + \dots + \overline{\mathbf{x}}\_{p} = n - 1, \ 0 \le \overline{\mathbf{x}}\_{i} \le n - 1 \\ & \qquad \qquad \qquad \qquad \qquad \prod\_{i=1}^{p} \rho^{\bullet}\_{(p-1)\overline{\mathbf{x}}\_{i} + 1} \{c\_{p,0}\}^{\overline{\mathbf{x}}\_{i}} ; o\_{l(\overline{\mathbf{x}}(i) + 1)} \cdots o\_{l(\overline{\mathbf{x}}(i) + (p-1)\overline{\mathbf{x}}\_{i} + 1)} \} \end{aligned} \right] = \text{const}$$

If 1 1 ( 1) 1 ( 1) 1 () () Re( )Im( )0 () () *p n p n Hj Hj LjLj* , then <sup>1</sup> ( 1) 1 ( ) ( ) *p n H j L j* has no effect, either. This gives

Equation (23). The proof is completed.

### **7. References**


[7] Jing X.J., Lang Z.Q., and Billings S. A., "Mapping from parametric characteristics to generalised frequency response functions of nonlinear systems", International Journal of Control, 81(7), 1071-1088, 2008a

32 Advances on Analysis and Control of Vibrations – Theory and Applications

1 ( 1) 1

*k k p n*

 

*c n*

( ()

1 ( 1) 1

*k k p n*

 

*n c*

( () ;

sgn

sgn

**7. References** 

*k*

*p n*

*L j*

*p n*

If 1 1

11, pp 1150-1160, 1985

Wiley, 1961.

( 1) 1 ( 1) 1 () () Re( )Im( )0 () () *p n p n Hj Hj LjLj*

Equation (23). The proof is completed.

*L j*

( 1) 1 all the different combinations

*p*

*j Hj <sup>n</sup>*

 

1

*i*

12 p 1p i

 

( 1) 1 ,0

*px p*

*i*

1 \*

 

*H j <sup>n</sup>*

*i*

( 1) 1 all the different combinations

1

*i*

, then <sup>1</sup>

Mathematical Society, AMS Chelsea Publishing, 1991

Laboratory of Electronics, Cambridge, Mass. Jul. 24, 1959.

Control, Vol. 79, No. 12, December, pp 1552–1564, 2006

*p*

of {x ,x ,...,x } satisfying x x 1, 0 x n-1

*c*

From Lemma 2, ( )*<sup>k</sup> j* has no effect on this equation. Then the equation above is equivalent to

12 p 1p i

*c*

( ) (x , ,x ) ( )

 

> of {x ,x ,...,x } satisfying x x 1, 0 x n-1

( 1) 1 ,0 ( ()

[1] Billings S.A. and Peyton-Jones J.C., "Mapping nonlinear integro-differential equation into the frequency domain", International Journal of Control, Vol 54, 863-879, 1990 [2] Boyd, S. and Chua L., "Fading memory and the problem of approximating nonlinear operators with Volterra series". IEEE Trans. On Circuits and Systems, Vol. CAS-32, No

[3] Bromwich T. J., An Introduction to the Theory of Infinite Series, American

[4] George D.A., "Continuous nonlinear systems", Technical Report 355, MIT Research

[5] Graham D. and McRuer D., Analysis of nonlinear control systems. New York; London :

[6] Jing X.J., Lang Z.Q., Billings S. A. and Tomlinson G. R., "The Parametric Characteristics of Frequency Response Functions for Nonlinear Systems", International Journal of

*i*

 

( 1) 1

( ) ( ) *p n H j L j*

*x px p lXi*

*i*

*x*

1 p

*const*

*const*

( ( ) 1) ( ( ) ( 1) 1)

1 p

1) ( ( ) ( 1) 1)

 

*lXi p x*

) *<sup>i</sup>*

has no effect, either. This gives

; ) *<sup>i</sup>*

*lXi lXi p x*

 

*x*

*x*

1 \*

() () (x , ,x ) ( )

	- [23] Jing X.J., Lang Z.Q., and Billings S. A., Determination of the analytical parametric relationship for output spectrum of Volterra systems based on its parametric characteristics, Journal of Mathematical Analysis and Application, 351, 694–706, 2009

**Chapter 0**

**Chapter 2**

**LPV Gain-Scheduled Observer-Based State**

**Disturbances with Time-Varying Frequencies**

The design of controllers for the rejection of multisine disturbances with time-varying frequencies is considered. The frequencies are assumed to be known. Such a control problem frequently arises in active noise and vibration control (ANC/AVC) applications where the disturbances are caused by imbalances due to rotating or oscillating masses or periodically fluctuating excitations, for example the torque of a combustion engine, and the rotational

For the rejection of disturbances with time-varying frequencies, time-varying controllers that are automatically adjusted to the disturbance frequencies are usually used. Although time-invariant controllers might be sufficient in some applications [22], time-varying controllers usually result in a much better performance, particularly if the disturbance frequencies vary over fairly wide ranges. Such a controller can be constructed in several ways (see Sec. 2.1). Two observer-based state-feedback controllers are presented in this chapter (see Sec. 3). General output-feedback controllers are treated in the next chapter. The approaches presented in this chapter use state augmentation in order to achieve disturbance rejection. One consists of a time-invariant plant observer and a time-varying state-feedback gain for the state-augmented plant, where the state augmenation is based on a time-varying error filter, as proposed by Kinney & de Callafon [19]. The other controller approach is based on the disturbance observer of Bohn et al. [7], where the plant is augmented with a time-varying disturbance model. A time-varying observer for the overall system and a time-invariant

The remainder of this chapter is organized as follows. In Sec. 2, existing approaches to the problem are classified and some general control design considerations are discussed. The state-augmented observer-based state-feedback approaches are described in Sec. 3. In Sec. 4,

> ©2012 Bohn et al., 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

©2012 Bohn et al., licensee InTech. This is a paper 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.

**Feedback for Active Control of Harmonic**

Wiebke Heins, Pablo Ballesteros, Xinyu Shu and Christian Bohn

speed is measured. Application examples are automobiles and aircrafts.

state-feedback gain are used to track and reject the disturbance.

cited.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50293

**1. Introduction**
