Vibration Control by Exploiting Nonlinear Influence in the Frequency Domain

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. 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. 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. …


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.
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.

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] , 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 c0,1(.) and c1,0(.) are referred to as linear parameters corresponding to coefficients of linear terms in the model, i.e., ( ) By using the probing method [16], a recursive algorithm for the computation of the nthorder generalized frequency response function (GFRF) for the NDE model (2) is provided in [1]. Therefore, the output spectrum of model (2) can be evaluated as [9] which is truncated at the largest order N and where, is known as the nth-order GFRF defined in [4], and is the nth-order Volterra kernel introduced in (1). When the system input is a multi-tone function described by F is the modulus, and K is a positive integer), the system output frequency response can be evaluated as [9]: where ) ( i k F  can be explicitly written as In order to explicitly reveal the relationship between model parameters and the frequency response functions above, the parametric characteristics of the GFRFs and output spectrum are studied in [6]. The nth-order GFRF can then be expressed into a more straightforward polynomial form as with terminating condition 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 where the terminating condition is k=0 and 1 (which is the transfer function when all nonlinear parameters are zero), represents the frequency variables involved in the corresponding functions, x is the number of the parameters in The mapping function  to be analytically and directly determined in terms of the first order GFRF and nonlinear parameters. Therefore, the nth-order GFRF can directly be written into a more straightforward and meaningful polynomial function in terms of the first order GFRF and model parameters by using the mapping function 1 ( ( ( )); , , ) . Similarly, Equation (6) can be written as 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.
Note that C1,1= c1,1, C0,2=c0,2, C2,0=c2,0. Thus, Proceed with the process above, the whole correlative function of 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 In this study, a novel property of the nonlinear influence on system output spectrum is revealed by using the new mapping function 1 ( ( ( )); , , ) 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.

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 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  . 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 Re( ( )) Re( ( )) Re( ( )) Re( ( )) (Im( ( )) Im( ( )) Im( ( )) Im( ( )) ) From definition 1, if ( ) Y j is an alternating series, then )) ( alternating. When (12) is an alternating series, there are some interesting properties summarized in Theorem 1. Denote 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 1 1 Re( ( )) Im( ( )) min , Re( ( )) Im( ( )) (12) has a radius of convergence R, the truncation error for a finite order  >T is , and for all n  0, is also an alternating series with respect to parameter c; Furthermore, 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 ( ) T Y j   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 Note that k0 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, and all the other parameters are zero.
Note that there is only one nonlinear term in the output in this case, the nth-order GFRF for system (15) can be derived according to the algorithm in [1], which can be recursively determined as Proceeding with the recursive computation above, it can be seen that , and substituting these equations above into (11) gives another polynomial for the output spectrum. By using the relationship (10) and the mapping function 1 ( ( ( )); , , ) , these results can be obtained directly as follows.
For simplicity, let ( ) 1, , l n   in (11b). By using (8) or Proposition 5 in [6], it can be obtained that Then the output spectrum at frequency  can be computed as (N is the largest order after truncated) ( ( ()); , , and    111)); The series is alternating. In order to check the series further, computation of 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.

Alternating conditions
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) ( 1 ) 1 can be computed from (11b), and n is a positive integer. Noting that ( ) 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 . In this case, If p is an even integer, then (p-1)n+1 is an odd integer for n=2k (k=1,2,3,…) and an even integer for n=2k-1 (k=1,2,3,…). When n is an odd integer, 1 ( This gives that ( 1) 1 ( ) =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 where const is a two-dimensional constant vector whose elements are +1, 0 or -1; (2) or if k1=k2=…=kp=k in cp,0(.), The recursive terminal of ( 1) 1 ,0 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 cp,0(k1,k2,…,kp) satisfying cp,0(.)>0 and 2 1 p r   for r=1,2,3,.... Similar results can also be established for any other nonlinear parameters. Regarding nonlinear parameter cp,0(k1,k2,…,kp) satisfying cp,0(.)>0 and 2 p r  for r=1,2,3,...., it can be obtained from (21a-c) that 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 cp,0(.)>0 with 2 p r  (even degree) than with respect to cp,0(.)>0 with 2 1 p r   (odd degree).
Note that Equation (21a) is based on the assumption that there is only nonlinear parameter cp,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 , p q C   includes all the nonlinear parameters in the system. Based on the parametric characteristic analysis in [6] and the new mapping function 1 ( ( ( )); , , ) [7], (24b) can be determined easily. For example, suppose p is an odd integer larger than 1, is given in (21c), and ( 1)  denotes a monomial consisting of some parameters in 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 The output spectrum at frequency  is given in (18)(19). From Lemma 3 in Appendix D, it can be derived for this case that Note that the terminal condition for (25b) is at i Therefore, from (25a-c) it can be easily shown that 2 1 3,0  represents the number of all the involved combinations which have the same . 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: which is a natural resonance frequency of model (15). It can be derived  denotes the odd integer not larger than n+1. Especially, when ( ) 1 Note that in all the combinations involved in the summation operator in (26) or condition (a2), i.e., This happens in the combinations where the argument of is either -90 0 or +90 0 . Note that there are more cases in which the arguments are -90 0 . If the argument is -180 0 , the absolute value of the corresponding imaginary part will be not more than 3  3  3  the combination  4  3  3  4  1  1 (1) ( ) whose argument is 0 0 ( ) -180 which is much less than Re( (x ,x ,...,x ) ) 0 ( )  (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.

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.   if the elements of C1 are the same as those of C2, 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, 1   Re( ( )) Re( ( )) Re( ( )) 1 Re( ( )) Re( ( )) Re( ( ))