Influence of excited and self-excited oscillations during the identification of non-linear systems.

Karic, Seniha ; Voloder, Avdo ; Baricak, Viktor 等

Abstract: In this study the influence between excited and self-excited oscillations during identification of a non-linear oscillating system using the phenomenological method was investigated. As the basis, the method uses the experimental results to create a mathematical model with one degree of freedom. The solution of the non-linear differential equation of the oscillatory motion is a sum of the harmonics, which are the functions of the exciting and own frequencies. In the study, the exciting force is considered as single or as a sum of more harmonic functions and accordingly the part of the solution coming from excitation is a function of one or more exciting frequencies. The expected increase of the amplitude after self-excitation shows the correctness of the method.

Key words: system identification, self-excited oscillations, excited oscillations, time domain, frequency domain.

1. INTRODUCTION

System identification is a widely used method in the technique to identify the model structure, estimate the model parameters and validate models. Identification of the non-linear dynamic systems--analoguously to the identification of the linear dynamic systems--can be classified into methods based on the analysis in time and frquency domain. It can be implemented with the correlation method, energy balancing method (Hadzic, 2004), harmonic balancing method (Woelfel, 2004), random modeling method etc. Despite that, the nonlinear system identification techniques can also be categorized as parametric and non-parametric. There exist the direct (Lacarbonara, 1997) and the indirect methods for the system identification. In this study the indirect or phenomenological method (Hadzic, 2004) for the system identification of a high speed cutter (Hadzic, 2004) is used.

Using the non-linear theory it is possible to solve the problems of determining the amplitude and the period of the oscillating mechanical systems, for the whole range of change of their parameters and characteristics, for which the corresponding mathematical model is valid (Rand, 2003). This enables the investigation of the global characteristics of the real mechanical systems, which can be used as the basis to specify the desired characteristics of the new-designed systems.

2. THE ANALYSIS OF EXPERIMENTAL DATA

For the system identification the analysis of experimental data in time and frequency domain (shown in Fig. 1) was utilized. The figure shows the behavior of a high-speed cutter during the working process characterized by presence of the self-excited oscillations caused by the interaction of the cutting tool and working part. Fig. 1 a) shows the system behavior in the time domain for the whole period of interest. A detailed view for the period before and after self-excitation is shown in Fig. 1c) and 1e) respectively. The diagrams in the frequency domain indicate that the increase of the amplitude is caused by self-exciting oscillations. Fig. 1.d shows the frequencies in the interval 0-10000 Hz before the amplitude increase (occurrence of self-excitation) and indicates that there exist only exciting frequencies. Fig. 1.b gives a more clear evidence about the dominant exciting frequencies before the self-excitation occur, while Fig. 1.f shows frequency domain after self-excitation in the interval between 0-10000 Hz, and indicates the existence a few of the own frequencies where frequency from 5100 Hz is dominant.

The exciting force has the harmonic character and can be represented in the general form as f=f(t+T)=f(t), i.e. decomposed into a number of harmonic functions (Woelfel, 2004). The analysis of exciting forces (Fig. 1.b) indicates that they are periodic and harmonic consisting of a number combined oscillations (Woelfel, 2004), with a basic circular frequency [[OMEGA].sub.1] = 2 x [pi] x 250 [s.sup.-1].

[FIGURE 1 OMITTED]

3. SYSTEM IDENTIFICATION

For the identification of exciting force and its influence on the self-exciting oscillations in the system, an analysis was carried out for individual frequencies of the exciting force of 250, 500 and 750 Hz, as well as for the combined exciting forces. Exciting force is defined by harmonic functions and following different cases will be considered:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, the amplitude of the exciting forces is taken as equal for individual exciting forces, which does not decrease the generality of the analysis. For combined exciting forces sum of the amplitudes is equal to the amplitude for singular analyzed exciting forces (F=[10.sup.-6]m, what is equal to the amplitude of experimental data before self-excitation). Different exciting forces are used to define a general mathematical model.

The motion before the occurrence of self-excited oscillations is described by the following linear equation:

[??] + 2 [omega]D[??] + [[omega].sup.2]q = 1/m [f.sub.j](t), j = 1, 2, 3, 4, 5, (2)

where, m is the mass of the system, d = 2[omega]Dm is the viscous dumping coefficient, k is the stiffness and [omega] is the frequency of the own oscillations. For the numerical solution of the differential equations the Runge-Kutta method of fourth order is used in this study. By introducing a corresponding non-linear term in the equation (2), the following non-linear equation is obtained:

m[??] + d[??] - [v.sub.0] x sign([??]) + kq = [f.sub.j(t), (3)

where, [v.sub.0] x sign([??]) 0 v is the non-linear term which represents the Coulomb's friction force (Guljajev, 1989), [v.sub.0] is constant representing the non-linearity coefficient.

Steady non-linear oscillations can be approximated by two components: one which represents the exciting oscillations with external exciting force, and another one which represents self-excited oscillations. So, in this case, the solution is not the sum of a homogeneous and a particular solution dependent on the excitatin frequency, like with linear oscillations (Vukojevic & Ekinovic, 2004), nor the harmonic function is dependent on the own frequency, like with free non-linear oscillations. The solution can be assumed as a sum of harmonic functions that contain the frequencies of exciting and own oscillations

q = Ccos([omega]t)+Q cos([[OMEGA].sub.i]t - [chi]), i = 1,2,3 (4)

where, Q is the amplitude of the periodic motion (before occurrence of self-excited oscillations), C is the amplitude in the steady state after the effect of the self-excitation, [chi] is the phase shift of excited oscillations. The non-linearity coefficient [v.sub.0] can be calculated using the method of energy balance (Hadzic, 2004), which is used for the transition area, when the oscillation amplitude increases for a given value. By applying the condition that C=2Q the results given in Table 1 are obtained. Method is general for every exciting force. Analytical and numerical results are corresponding.

The results in time and frequency domain for different exciting forces are shown in Fig. 2. The analysis shows the corresponding increase of the amplitude (time domain), which means that the resulting steady oscillations are self-excited. For different exciting forces the amplitude increase is obtained. In frequency domain there is appearance of own frequency after self-excitation as experimental data shows.

4. CONCLUSION

This analysis has shown that the non-linearity of the exciting force is of less importance for the given case then the self-exciting force, and it can be approximated with only one harmonic function even for the case that the exciting force consists of more excitations (lower, higher, combined) without decreasing the accuracy of the mathematical model. The non-linearity is caused mainly by self-excited oscillations. It is important to define a good ratio between the amplitudes before and after the self-excitation (in time domain) in order to define the corresponding non-linearity coefficient, which is in this case the dry friction coefficient. This would not be possible without using a method for the analysis of the non-linear systems, like the energy-balance method. Important basis for the system identification using the phenomenological method is the accurate identification of exciting and own frequencies in the frequency domain. Future research will be focused on the extension of the method to the systems with two degrees of freedom.

[FIGURE 2 OMITTED]

5. REFERENCES

Guljajev, V.I.; Bazenov, V.A. & Popov, S.L. (1989), Appropriate exercises of the theory of the non-linear oscillations of mechanical systems, Moscow (in Russian).

Hadzic, S. (2004). Contribution to the phenomenological approach of the mathematical modeling of oscillations of mechanical systems, Master thesis, Mechanical engineering faculty, University of Tuzla (in Bosnian).

Lacarbonara, W. (1997). A Theoretical and Experimental Investigation of Nonlinear Vibrations of Buckled Beams, Blacksburg, Virginia

Rand, R. H. (2003). Lecture Notes on Nonlinear Vibrations, Dept. Theoretical & Applied Mechanics, Cornell University.

Vukojevic, D. & Ekinovic, E. (2004), Theory of oscillations, Mechanical engineering faculty Zenica, University of Sarajevo (in Bosnian).

Woelfel, H. P. (2004). Mashine dynamics, Darmstadt (in German).

Table 1. Amplitudes before and after self-excitation for
different exciting forces, analytical and numerical results.

 Analitical Numerical Analitical
Excitation solution solution solution
force [Q10.sup.-6] (m) [Q10.sup.-6] (m) [C10.sup.-6] (m)

[f.sub.1] (t) 1.00 1.00 2.00
[f.sub.2] (t) 1.01 1.01 2.02
[f.sub.3] (t) 1.02 1.01 2.04
[f.sub.4] (t) 1.01 1.01 2.01
[f.sub.5] (t) 1.01 1.05 2.02

 Analitical Numerical Numerical
 solution solution solution
Excitation (Q + C) [A.sub.min] [A.sub.min]
force [10.sup.-6] (m) [10.sup.-6] (m) [10.sup.-6] (m)

[f.sub.1] (t) 3.00 0.95 3.00
[f.sub.2] (t) 3.03 1.01 3.03
[f.sub.3] (t) 3.06 0.91 3.05
[f.sub.4] (t) 3.02 1.06 3.01
[f.sub.5] (t) 3.03 0.95 3.02

Excitation
force [v.sub.0]m

[f.sub.1] (t) 32.34
[f.sub.2] (t) 32.57
[f.sub.3] (t) 32.97
[f.sub.4] (t) 32.45
[f.sub.5] (t) 32.58