for                                                              (1.3)


     


 


 


 


In the equation above, denotes the  n th order output frequency response of the system. This expression reveals


 


how the non-linear mechanisms operate on the input signal to produce output response in frequency domain.


 



                                                                                                                                           (1.4)
 


is the definition of the nth-order Generalized Frequency Response Function (GFRF)


 


 


(Billings and Yusof 1996, Boaghe  et al . 2000, 2002) , and


 


 



 


denotes the integration of          over the n-dimensional hyper plane    Equation (1.3) extends the well-known linear relationship (1.1)  to the non-linear case.


 


From equation (1.1), it shows that the possible output frequencies are exactly the same as the frequencies in the input for


 


linear system. For equation (1.2), the situation of non-linear system becomes more complicated than that in linear case,


 


that is, the relationship between input and output frequencies is generally given by


 


                          (1.5)


 


 


Where represents the non-negative frequency ranges of the system output, and denotes the non-negative frequency


 


ranges produced by the  n th-order system non-linearity. 


 


In 1997, Lang and Billings derived an explicit expression for the output frequency range    of non-linear systems when


 


subjected to a general input with a spectrum given by


 


 


                                           when



                     0          otherwise


 


where b>a The result obtained is                   


 



 


 



 


   


 


                                                     (1.7)


        


     means to take the integer part


               for k=0,…….,   


 


 


 


 


In (1.7) p* could be taken as 1,2,…,   , the specific value of which depends on the system nonlinearities. If the  


 


system GFRFs     for i=1,…,q-1, and   , then p*=q. This is the first analytical description


 


for the output  frequencies of non-linear systems, which is an extension of the relationship between the input and output


 


frequencies of linear systems in equation (1.1) to the non-linear case.


 


 


The theory above shows the basic properties of the relationship between the input and output frequencies. However, more


 


details about the relocation of the frequency domain response taking place of a non-linear system can not be acquired.


 


Therefore, a new frequency domain analyse method of non-linear system which is known as


 


NOFRFs is introduced in the next section.


 


 


 


 


 


 Charpter 2   Non-linear output frequency response functions (NOFRFs)


 


2.1  Some basic concept


 


First consider two different systems in time domain Static linear system


 


   y(t) =ku(t)


 


Static non-linear system


 


y (t)=ku(t)


 


The frequency domain representations of the systems above are


 


            Y(jw)= kU(jw)


 


and


 


     Y(jw)= kU(jw)


 


                 respectively.  U(jw)  represents the Fourier transform of u(t)  and it can be written  


as


 


                                         (2.1)
 


 


The equation (2.1) should be taken as an expression of a second order non-linear system.


 


In this case, a more generalized expression can be deduced



 


 


 


 


          


 


Equation (2.2) is the Fourier transform of the system input u(t) raised to the  n th power. From the concepts above, the


 


expression of output frequency response of a stable time invariant linear system described in (1) can be rewritten as


 


Y(jw)=G(jw)U(jw)


 


Where   Y(jw)= Y (jw), G(jw)= G (jw), U(jw)= U (jw)


 


            Under the definition that  Un(jw)is the natural extension of  U(jw) to the  n th order homogeneous non-linear situation, equation (3) can be rewritten as


 


 


= G(jw)U (jw)


 


 


 


under the condition


 



 


 


The concept of NOFRFs is defined as        


 


 



 


 


 


which was introduced by Lang and Billings [1]


 


One of the important properties of NOFRFs is


 


Y (jw)= Y (jw) = G(jw)U (jw)


2.2  Evaluation of  G(jw)


 


In this section, a direct evaluation of  G(jw)will be educed from the definition in  (2.5).


 


From equation (2.6), it is known that


 


Y (jw) = G(jw)U (jw)=  


 


=


 


in which   and        are the real and imaginary part of  ,  and   are the real and imaginary part of  . Therefore


 



 


 


             in which   and 


 


              Consider the case of   where  u(t)=u(t) where  is a constant and u(t)   is the input signal.


 


The equation above can be written as 


 



 


 


 


           in which  and  are the real and imaginary part of    ,  and are the real and imaginary part of  G *.


 


Stimulate the system with input signals


 


           where


From NOFRFs, the output frequency response under    input can be expressed as below.


 


 



 


 


where


 


 



and


 


 


 


 



 


 


 


 


 


 


 


 


Using a least square based approach, the expression of  G * can be written as  


 


 


                             (2.7)
 


           =


 


 


The requirement of this algorithm for the determination of the non-linear output frequency response is experimental or


 


simulation data of the system under different input excitations. In the following chapter, this algorithm will be applied


 


to an example non-linear system to evaluate its NOFRFs.


 


 


Charpter 3  NOFRFs based analysis for a mechanical oscillator


 


 


3.1  Introduction of the non-linear system


 


In this section, the procedure of quantitatively calculates the NOFRFs to investigate the generation of new frequency


 


components of output signal in non-linear system will be applied to a non-linear system.


 


The system which can be described by the non-linear system is shown below


 



 


 


 



 


 


 


 


 


 


 


 


Figure 1. A mechanical oscillator


 


the differential equation of this system can be described as


 


                                       (3.1)
 


 


 


where u(t) and y(t) are the input and output of the system. and   are the system parameters. In this case, assuming m =1, c =20,  ,   , , to study the behaviour of the non-linear system in frequency domain when subject to a specific input using the analysis procedure in §1.3


 


                                 


 


 3.2  Analysis of system output frequency response 


 


The specific input is given by


 


                      (3.2)  


 


       t  =  -511 0.005 sec, …, 512 0.005 sec


 


Having been given the input signal, the output can be obtained in MATLAB by solving ordinary difference equation.


 


MATLAB uses Runge-Kutta-Fehlberg method solving ordinary difference equations (ODE). The result is calculated


 


within limited sampling point. In this case, the MATLAB function ode45 will be used to calculate the time domain output


 


signal which is shown in Figure 3 (b). The code is shown in Appendix A.  Figure 4 shows the spectra of input and


 


corresponding output signal using Fast Fourier Transform from the time domain input and output signals, the MATLAB


 


code is shown in Appendix B. From Figure 4, it is very clear that the output spectra have richer frequency components


 


than those of input signal. Which indicates that some of the input energy is transferred by the system from the input


 


frequency band (30,55) Hz to a lower frequency range (0,30) Hz. This phenomenon is a very significant property of non-


 


linear systems. However, further study on the properties of this non-linear system will depends on the investigation of


 


non-linear output frequency response corresponding to input (3.2).


 


 


 


 


 


 


 


 


const          constant          0 


   


                                        x=       y               output           1000 


u                input             2000 


   e                noise             3000 


 


subsystem  =up to 9 subsystems are allowed = number from 1 to 9 times 100


 


lag        number from 1 to 99   


 


The maximum power that a term can be raised to is 4.


 


Using the code convention described above, the NARMAX models can be represented as the matrix showing in Appendix F.


 


Given an input signal, the corresponding output response can be generated using MATLAB in SUN workstation.


 


The signal (3.1) is chosen as the input signal to the three models


 



 


 


The output responses corresponding to input signal (4.1) of the three given non-linear


 


systems are generated in laboratory using Sun workstation, and the code is shown in


 


Appendix G. The input and corresponding output signal of the three studied systems


 


are showing below:


 


   


 


                                                     (1.7)


        


     means to take the integer part


               for k=0,…….,   


 


 


 


From (4.1), it is easy to find out that the corresponding frequency range of  G(jw) n=1,2,3, is (30,55) Hz,(60,100) Hz and


 


(50,800) Hz respectively.


 


First, considering the first order homogeneous non-linear system, the first order transform functions over a frequency


 


range (30, 55) Hz of each model are generated and shown in figure 18 in the form of spectrum.


 





 


                                


 


                                                 Figure 19. The first-order non-linear output transfer functions of three models


 


 


From Figure 19, it can be clearly observed that the non-linear output transfer function of the first model has the minimal


 


gain comparing with other two models while the third model holds a maximum gain.


 


The following non-linear output transfer functions belong to the second order homogenous non-linear system and the


 


NOFRFs over a frequency range of (60,100) Hz are showing below:


 


 


 





 


 


 


 


Figure 20. The second-order non-linear output transfer functions of three models


 


Obviously, the NOFRFs that are shown in Figure 20 share the same trend with Figure 19. of model 1 has a maximum value around  1.5 in frequency range (60,100) Hz while the other two models hold a maximum value of 0.03 and 400 respectively.


 


Then, the non-linear output transfer functions of the third order non-linear


 


components of these three models in frequency range (5,80) Hz are evaluated and


 


shown in Figure 21:





 


 


 


 


 


 


                                               Figure 21. The third-order non-linear output transfer functions of three models


 


Again, the same phenomenon is presented in Figure 21 that is exactly the same with previous two homogeneous non-


 


linear systems.


 


For more details of the comparison of these three models, the integrals of each G(jw) over the investigated frequency


 


ranges are evaluated and which are shown in the table 3. From the table below, the different between each model is very


 


evident. 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 


 Table 3 compare of integral of   for each model


 


 


 


4.4  Conclusion


 


In this chapter, the concept of NOFRFs is applied to a real mechanical structure andthe output transfer function up to


 


fourth order homogeneous non-linear systems are evaluated which are shown in figure 18, figure 19 and figure 20. From


 


above paragraphs, it is able to compare non-linear output transfer function of each model up


 


to third order in specified frequency range. The result shows a direct relationship between the faults of non-linear system


 


and the size of output transfer functions. From Table 3, it is even more evident that model 3 always holds the largest


 


magnitude of G(jw)   than other models in each order. It is also known that the first model is a faultless system while


 


model 3 has the most significant fault. The study in different order of  G(jw) and various frequency ranges all indicate


 


that the more faulty systems are, the larger the magnitudes of output transfer function. This conclusion indicates that the


 


algorithm of NOFRFs can be used in fault detection of non-linear systems to identify the size of faults of non-linear


 


systems.


 


 


 


Charpter 5  Conclusion


 


 


In this paper, a methodology for fault detection in non-linear systems based on the concept of NOFRF (Non-linear Output


 


Frequency Response Function) has been introduced, and been applied to two different types of non-linear models.


 


In chapter 2, the first model is given in ODE (Ordinary Differential Equation) form. With a given input signal in time


 


domain, the corresponding output response is directly generated in MATLAB using function ode45. The NOFRFs are


 


then calculated using the approach introduced in Chapter 1 and the validity of this approach also been validated.


 


The second system is given in NARMAX model in chapter 3. A set of codes are written to obtain its output corresponding


 


to given input in SUN workstation. In this case, three Aluminium plates with different type of fault are introduced in


 


NARMAX model form. By studying and comparing the NOFRFs of each plate in certain frequency band, it is easy to


 


discover the relationship between the types of fault and size of  . In this way, it can be found that NOFRFs-based


 


approach is an effective method in the application of fault detection. 


 


 



 


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