Figure 1. Free vibrations of a bilinear system with single potential well. (a) Bilinear characteristic; (b) back-bone curve; (c) free damped oscillations, where Ψ – loss coefficient. Parameters: c1 = 50, c2 = 1, d = 0.01, b = 0.5.

(b)

(c) b = 0.5, Ψ = 0.81

с1

с2

d

(a)

Complex Nonlinear Effects and Rare Attractors in Bilinear Valve Systems Under Forced

Excitations E.P. Shilvan, M.V. Zakrzhevsky

Institute of Mechanics, Riga Technical University, Riga, Latvia e-mail: ed_shilvan@inbox.lv, mzakr@latnet.lv

Abstract— Bilinear systems are widely used in different

engineering systems and incomplete research of these systems can lead to disaster. The main task of this paper is to study the dynamics of valve system by conducting complete bifurcation analysis, using the method of complete bifurcation groups (MCBG).

I. INTRODUCTION Nonlinear vibration models with bilinear elastic

characteristic is widely used in research of forced oscillations in different applications: for analysis of the dynamics of offshore structures, valves and switches, in the models of suspension bridges, in the description of the dynamics cracks, for systems with soft impact [1-3].

This article discusses the dynamics of a simple bilinear valve system under forced excitation. For the study of this system so-called bifurcation theory of nonlinear dynamical systems and the method of complete bifurcation groups were used [4-7]. The aim of the research is an interaction of two different 1T and 2T bifurcation groups by changing the frequency and amplitude of excitation force. The stiffness ratio c1/c2=50 is considered.

II. DYNAMICAL MODEL

The simplest bilinear system with linear dissipation and harmonic excitation force is considered. The equation of motion for valve mass in nondimensional form is such:

thxfxbxm ωcos)( =++ . (1)

( )⎩⎨⎧

≥−−<

=dxifdccxcdxifxc

xf,

,)(

122

1 (2)

where m – mass of oscillating valve; b - coefficient of linear dissipation; c1 and c2 - restoring force coefficients c1 >> c2; d - break point of bilinear characteristics; h, ω - the amplitude and frequency of the excitation.

Bilinear characteristic, back-bone curve and free damped oscillations are shown in Fig. 1. Nondimensional parameters of the studied model are: c1 = 50, c2 = 1, d = 0.01, b = 0.5, ω = var. and h = var.

III. RESULTS

For the base parameters of the bilinear valve system bifurcation diagrams were built by changing the frequency ω and amplitude h of excitation force. All results were

obtained numerically, using software NLO [8] and SPRING [9], created in Riga Technical University.

Figures 2-3 display the coordinates of the fixed points of stable orbits and the amplitude of the oscillations. Figure 2 (a) shows that at low values of the amplitude h = 0.1 the system has only one bifurcation group 1T, but with increasing amplitude up to h = 0.5 (Fig. 2,b) there is a new group – subharmonic isle 2T. Increasing the amplitude of the driving force is accompanied by a change in the structure of the bifurcation of 1T, namely the appearance of period doubling bifurcations that later leads to an infinite number of unstable periodic orbits (unstable periodic infinitium - UPI). In Fig. 3(e) for h = 1 it is shown that the two bifurcation groups 1T and 2T join together into a complex group with protuberance. The subsequent increase of the amplitude to h = 2 leads to the birth of rare attractors (RA) and new regions of parameters with UPI (Fig. 3,f-h).

NSC 2012 – 4th IEEE International Conference on Nonlinear Science and Complexity • August 6-11, 2012 • Budapest, Hungary

217978-1-4673-2703-9/12/$31.00 ©2012 IEEE

Figure 2. Forced vibrations of a bilinear system with linear dissipation and periodic harmonic excitation. Two bifurcation groups 1T and 2T are shown, which join together to the complex group (continuation in Fig. 3) by increasing the amplitude of the excitation force. There are also subharmonic bif. groups 3T, 5T, 6T. In left column the coordinate x of fixed points and in right column amplitude of oscillations are shown. Stable solutions are plotted by solid lines and unstable - by thin lines (reddish online). Parameters: c1 = 50, c2 = 1, d = 0.01, b = 0.5, h = var., ω = var.

1T

h = 0.1,

1T

2T h = 0.5

2T

1TP2

h = 0.75

2T

1TP2

h = 0.95

1T2T

h = 0.5

1T

h = 0.1,

P1

1T

2T P2

h = 0.95

P1 1T

2T P2

h = 0.75

ω = var. ω = var.(a)

(d)

(c)

(b)

E. P. Shilvan, M. V. Zakrzhevsky • Complex Nonlinear Effects and Rare Attractors in Bilinear Valve Systems Under Forced Excitations

218

Figure 3. (continuation of Fig. 2) Forced vibrations of bilinear system with linear dissipation and periodic harmonic excitation. Two bifurcation groups 1T and 2T are shown, which join together to the complex group by increasing the amplitude of the excitation force. There are also subharmonic bif. groups 3T, 5T, 6T. In left column the coordinate x of fixed points and in right column amplitude of oscillations are shown. Stable solutions are plotted by solid lines and unstable - by thin lines (reddish online). Parameters: c1 = 50, c2 = 1, d = 0.01, b = 0.5, h = var., ω = var.

P2

P2 P2 RA

P4 P2

h = 1.25

P2

P2 RA

P4

UPI

P4

P1 P2

h = 2.0

P2

P2 P2

h = 1.0

P2

P1P2 P4

P4 P4

P2 RA UPI

h = 1.5

P2 P2

P2

P4P2 RA

UPI

h = 2.0

P2 P2

P1

h = 1.0

P2 P2

P2 RA P4

P1 P2 UPI

h = 1.5

h = 1.25

P2 P2

P2 RA P4

P2

ω = var. ω = var. (e)

(h)

(g)

(f)

NSC 2012 – 4th IEEE International Conference on Nonlinear Science and Complexity • August 6-11, 2012 • Budapest, Hungary

219

Figure 4. Forced vibrations of bilinear system with linear dissipation and periodic harmonic excitation. Bifurcation groups of 1T, 3T, 5T and 6T are shown. Coordinates of fixed point (displacement and velocity) and amplitude of oscillations are shown. Stable solutions are plotted by solid lines and unstable - by thin lines (reddish online). Parameters: c1 = 50, c2 = 1, d = 0.01, b = 0.5, h = var., ω = 10

P1

P2

3T

5T6T

P1

P2

3T

5T

6T

ω = 10,

P1

P2

3T

5T

6T

(a)

(b)

(c)

h = var.

E. P. Shilvan, M. V. Zakrzhevsky • Complex Nonlinear Effects and Rare Attractors in Bilinear Valve Systems Under Forced Excitations

220

Figure 5. Forced vibrations of bilinear system with linear dissipation and periodic harmonic excitation. (a) - Basins of attraction for two attractors P2; (b) – phase projection for regime P2(2/2); (c) – phase projection for regime P2(1/2). Parameters: c1 = 50, c2 = 1, d = 0.01, b = 0.5, h = 0.95, ω = 5.1

h = 0.95,

P2 (1/2)

P2 (1/2)

P2 (2/2)

(a)

(b) P2 (2/2) (c) P2 (1/2)

ω = 5.1.

Figure 6. Forced vibrations of bilinear system with linear dissipation and periodic harmonic excitation. (a) - Basins of attraction for two attractors P2; (b) – the phase projection for regime P2(2/2); (c) – the phase projection for regime P2(1/2). Parameters: c1 = 50, c2 = 1, d = 0.01, b = 0.5, h = 0.95, ω = 9.5

h = 0.95,

P2 (1/2)

P2 (1/2)

P2 (2/2)

(a)

(b) P2 (2/2) (c) P2 (1/2)

ω = 9.5.

NSC 2012 – 4th IEEE International Conference on Nonlinear Science and Complexity • August 6-11, 2012 • Budapest, Hungary

221

For the considered system the bifurcation diagram is constructed for ω = 10 and h = var. In addition to the main 1T bifurcation group, there are subharmonic groups 3T, 5T and 6T with greater amplitudes of oscillations (Fig.4).

For values of frequency ω = 5.1 and ω = 9.5 of the excitation force basins of attraction for two coexisting regimes of period-2 are built (Fig.5,6). Figure 6 shows that the basin of attraction of orbit P2 (2/2) is significantly reduced at ω = 9.5 compared with one at ω = 5.1. Periodic orbit P2 (1/2) becomes globally stable after joining together of 1T and 2T bifurcation groups.

CONCLUSIONS It is shown that the method of complete bifurcation

groups allows conducting the global bifurcation analysis of the valve system with harmonic excitation, and finding new bifurcation groups with rare attractors.

The paper shows how the structure of the bifurcation diagram is changing by varying the parameters of the excitation force, e.g. two different bifurcation groups join together in one complex group.

Also it is shown that varying the parameters of the excitation force leads to new nonlinear effects, e.g. rare attractors and UPI.

ACKNOWLEDGMENT This work has been supported by the Latvian Council

of Science under the grant 09.1587 for year 2012.

REFERENCES [1] Thompson, J.M.T. and Stewart, H.B. Nonlinear Dynamics and

Chaos, 2nd ed., Wiley, 2002. [2] Beresnevich V., Tsyfansky S., Nonlinear Vibration Analysis of

Machine Oil Viscosity. In Proceedings of IUTAM/IFToMM Symposium on Synthesis of Nonlinear Dynamical Systems, 2000. p.85-90.

[3] Yevstignejev. V.Yu., Application of the Complete Bifurcation Groups Method for Analysis of Strongly Nonlinear Oscillators and Vibro-Impact Systems, PhD thesis, Riga, 2008. [in Russian].

[4] Zakrzhevsky M.V. Global Nonlinear Dynamics Based on the Method of Complete Bifurcation Groups and Rare Attractors, Proceedings of the ASME 2009 (IDETC/CIE 2009), San Diego, USA, 2009, p. 8.

[5] Zakrzhevsky M.V., Rare attractors and the method of complete bifurcation groups in nonlinear dynamics and the theory of catastrophes. Proceedings of 10th Conference on Dynamical Systems - Theory and Applications "DSTA 2009", Łódź, Poland, December 7-10, 2009, Volume 2, pp. 671-678.

[6] Zakrzhevsky M., Bifurcation Theory of Nonlinear Dynamics and Chaos. Periodic Skeletons and Rare Attractors. // Proceedings of the 2nd International Symposium RA'11 on "Rare Attractors and Rare Phenomena in Nonlinear Dynamics", May 17 - 20, 2011, Riga-Jurmala, Latvia, pp. 1-10.

[7] Zakrzhevsky M.V., Smirnova R.S., Schukin I.T., Yevstignejev V.Yu., Frolov V.Yu., Klokov A.V., Shilvan E.P. Nonlinear dynamics and chaos. Bifurcation groups and rare attractors. – Riga: RTU, Publishing House, 2012. – 181 p. [in Russian].

[8] Zakrzhevsky M., Ivanov Yu., Frolov V. NLO: Universal Software for Global Analysis of Nonlinear Dynamics and Chaos. In Proceeding of the 2nd European Nonlinear Oscillations Conference, Prague 1996. v.2., p. 261-264.

[9] Schukin I. Development of the methods and algorithms of simulation of nonlinear dynamics problems. Bifurcations, chaos and rare attractors. // PhD Thesis, Riga - Daugavpils, Latvia (2005), 205 p. [in Russian].

E. P. Shilvan, M. V. Zakrzhevsky • Complex Nonlinear Effects and Rare Attractors in Bilinear Valve Systems Under Forced Excitations

222