EXPERIMENTAL INVESTIGATION OF A TWO-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEM

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 22, No. 5 (2012) (7 pages) c World Scientific Publishing Company DOI: /S EXPERIMENTAL INVESTIGATION OF A TWO-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEM GUILIN WEN, HUIDONG XU, LU XIAO, XIAOPING XIE, ZHONG CHEN and XIN WU State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Key Laboratory of Advanced Design and Simulation Techniques for Special Equipment, MoE, Hunan University, Changsha, Hunan , P. R. China wenguilin@yahoo.com.cn Received January 23, 2011; Revised March 6, 2011 Vibro-impact systems with intermittent contacts are strongly nonlinear. The discontinuity of impact can give rise to rich nonlinear dynamic behaviors and bring forth challenges in the modeling and analysis of this type of nonsmooth systems. The dynamical behavior of a two-degreeof-freedom vibro-impact system is investigated experimentally in this paper. The experimental apparatus is composed of two spring-linked oscillators moving on a lead rail. One of the two oscillators connected to an excitation system intermittently impacts with a spherical obstacle fixed on the thick steel wall. With different gap sizes between the impacting oscillator and the obstacle, the dynamical behaviors are investigated by changing the excitation frequencies. The experimental results show periodic, grazing and chaotic dynamical behaviors of the vibro-impact system. Keywords: Experiment; vibro-impact; nonsmooth dynamics; impact oscillator. 1. Introduction Vibro-impact phenomena exist widely in practical mechanical systems such as impact print hammers, loaded spur gears, robot manipulation components and linkage joints with contact disengage process. The strong nonlinear characteristic caused by instantaneity and discontinuity of impact can exhibit rich dynamic behavior. Periodic motion, resonance, bifurcation and chaos of vibro-impact systems have been investigated extensively by theoretical and numerical analyses [Shaw & Holmes, 1983; Nordmark, 1991; Foale & Bishop, 1994; Luo & Xie, 1998; Wagg & Bishop, 2001; Wen et al., 2004; Yu & Luo, 2004; Xie & Ding, 2005; Kryzhevich, 2007; Yue & Xie, 2008; Rong et al., 2010; Li et al., 2011; Gan & Lei, 2011]. In particular, nonsmooth bifurcation with a reflection of the discontinuous nature of vibro-impact systems such as grazing bifurcation and sliding bifurcation have been exhaustively studied in recent years in [Luo & Chen, 2005; Wagg, 2005; Thota & Dankowicz, 2006; Budd & Piiroinen, 2006; di Bernardo et al., 2008; Nordmark et al., 2009; Pavlovskaia et al., 2010]. In addition, Wang et al. [2008], Tornambe [1999] and Wen et al. [2010] addressed the control problem of nonlinear dynamics of vibro-impact systems. Some researchers have experimentally investigated the dynamic behavior of vibro-impact systems. Stensson and Nordmark [1994] conducted an experimental investigation on chaotic dynamics of a mechanical system with low velocity impacts. Oestreich et al. [1997] investigated the dynamical behavior of a single-degree-of-freedom impact oscillator by experiments, comparing to simulations

2 G. Wen et al. Divenyi et al. [2008] showed the rich dynamical behavior of an oscillator with discontinuous support by experiments. Chicone et al. [2009] discovered the nonmonotonicity of the period function for a benchtop impact oscillator that incorporates Hertzian contact and experimentally verified a consequence of such behavior. Wilcox et al. [2009] investigated the nonsmooth fold bifurcations associated with a near-grazing bifurcation in a vibro-impacting mechanical system by analytical, numerical and experimental methods. The experimental investigation and the mathematical modeling of the impact forcebehaviorinanembeddedvibro-impactsystem were done by Aguiar and Weber [2011]. Nguyen et al. [2008] presented the experimental study and mathematical modeling of a vibro-impact moling rig based on electro-mechanical interactions of a conductor with an oscillating magnetic field. This paper deals with the dynamic behavior of a two-degree-of-freedom vibro-impact system via experiment. The design of the testing rig and experimental procedures are described in detail. The rich dynamical behaviors for the different gap conditions are revealed by changing excitation frequencies. The experimental study in this paper is attempted to lay the foundation for experimental investigation on anti-control of bifurcations [Chen et al., 2001; Wen et al., 2010] of vibro-impact systems in future work. 2. Experimental Rig Figure 1 is the schematic diagram of the experimental rig for a two-degree-of-freedom vibro-impact system. The nonlinear dynamics of the simplified model of this vibro-impact system have attracted much interest in theoretical studies [Luo & Xie, 1998; Wen et al., 2004]. As shown by the photograph in Fig. 2(a), the experimental apparatus is composed of two spring-linked oscillators moving on lead rail (label 5) along the horizontal direction. The steel mass (label 1), which is rigidly connected with the connecting rod of an electromagnetic exciter (label 6), intermittently impacts with a spherical obstacle (label 7) fixed on the thick steel wall (label 8). The steel masses (labels 1 and 2) are connected by the spring (label 3). The steel mass (label 2) is connected to the rear wall (label 9) by the spring (label 4). The different gap sizes can be obtained by adjusting the location of the rear wall (label 9) on the support base. With the different Fig. 1. Schematic diagram of the experimental rig. (a) Fig. 2. Photograph of (a) experimental rig, (b) laser sensor system and (c) real-time display

3 Experimental Investigation of a Two-Degree-of-Freedom Vibro-Impact System (b) (c) Fig. 2. (Continued) gap sizes between the impacting oscillator and the obstacle, the dynamical behaviors are investigated by changing the excitation frequencies. During the process of the steel mass (label 1) impacting with the spherical obstacle (label 7) on the thick steel wall (label 8), the displacement and speed of the Table 1. Specifications of the devices. Laser Sensor-V1002 Frequency range Sensitivity of displacement Sensitivity of velocity Output voltage Low-pass filter High-pass filter Noise reduction 0.1 Hz 3MHz 1 V/mm 1 V/m/s ±10 V 25 khz 0.1 Hz Low Electromagnetic Exciter-HEV-2 Maximum exciting force 200 N Static constant 7.5 N/A Maximum amplitude ±10 mm Maximum permitting peak current 28 A Permitting frequency range 0 2 (KHz) Power amplifier 550 W steel mass (label 1) are measured by the laser sensor system [label 10 and Fig. 2(b)]. The real-time display of the experiment is shown in Fig. 2(c) and the specifications of the devices on the test rig are listed in Table Experimental Results In this section, the dynamical responses of the vibro-impact system are investigated by changing the excitation frequency and the gap size between the steel mass (label 1) and the spherical obstacle (label 7). In order to wipe off efficiently the measuring noise and interference, the digital filtering of Laser sensor-v1002 (see Table 1) is used to recover the dynamical responses of the vibro-impact system. We choose physical parameters M 1 =14.95 kg, M 2 = kg, K 1 = N/mm, K 2 = N/mm. Let T, T e and ω denote the period of solution, the period of exciting force and the exciting frequency, respectively. When the gap is set at the range of 1 mm 3 mm, at the exciting frequency ω =7.4 Hz, the system response in Fig. 3(a)

4 G. Wen et al. (a) (b) (c) (d) Fig. 3. Periodic motions on time response, frequency spectrum and phase portrait. (a) ω =7.4Hz, T =0.135 s, T e =0.135 s with T = T e,(b)ω =19.56 Hz, T =0.102 s, T e =0.051 s with T =2T e,(c)ω =20.5Hz, T =0.147 s, T e =0.049 s with T =3T e,(d)ω =39Hz,T = s, T e = s with T =4T e and (e) ω =32Hz,T = s, T e = s with T =6T e

5 Experimental Investigation of a Two-Degree-of-Freedom Vibro-Impact System (e) Fig. 3. is related to a periodic motion with period T = T e, which indicates an impact per exciting period. When ω increases to Hz, the system gives rise to a periodic motion with T =2T e for an impact per two exciting periods [see Fig. 3(b)]. With further increase of ω up to ω =20.5 Hz, a periodic motion with T =3T e for an impact per three exciting periods appears in the system [see Fig. 3(c)]. When the gap is adjusted under 1 mm, at higher exciting frequency ω = 39 Hz, a periodic motion for an impact per four exciting periods with T =4T e appears [see Fig. 3(d)]. When ω decreases to 32 Hz, a periodic (Continued) motion with T =6T e for an impact per six exciting periods arises from the system [see Fig. 3(e)]. We next discuss a special situation when an impact with zero velocity occurs, namely, grazing impacts. When the gap is adjusted in 0 mm 1 mm and the exciting frequency is set at ω =10.9Hz, there exits a periodic motion whose internal section does not touch the collision boundary B [see Fig. 4(a1)]. With the increase of ω up to ω = 11.1 Hz, the internal section of the periodic motion grazes the boundary B and the grazing phenomenon of zero-velocity impact happens in (a1) ω =10.9Hz (b1) ω =11.1Hz (c1) ω =11.2Hz (a2) ω =10.9Hz (b2) ω =11.1Hz (c2) ω =11.2Hz Fig. 4. Grazing phenomenon

6 G. Wen et al. Fig. 5. Chaos motions at ω =45Hz. the vibro-impact system [see Fig. 4(b1)]. When ω increases to 11.2 Hz, the internal section of the periodic motion impacts with the boundary B [see Fig. 4(c1)]. As shown in the time response diagrams in Figs. 4(a2) 4(c2), the grazing phenomenon is further shown by analyzing the relations between the displacement x and the boundary B. In addition, with the increase of the exciting frequency ω to ω = 45 Hz, the chaos motion with continuous spectrum appears in the vibro-impact system (see Fig. 5). 4. Conclusions and Remarks In this paper, an experimental apparatus has been built to investigate the dynamical behaviors of a two-degree-of-freedom vibro-impact system. The experimental results show the dynamic responses of periodic motion, grazing impact and chaotic motion of the vibro-impact system by changing the excitation frequencies and gap sizes. The experimental rig will be developed to investigate quasiperiod motions with strong parameter-dependence and anti-control of bifurcations of vibro-impact systems in future work. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos ; ) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No ). References Aguiar, R. R. & Weber, H. I. [2011] Mathematical modeling and experimental investigation of an embedded vibro-impact system, Nonlin. Dyn. 65, Budd, C. J. & Piiroinen, P. T. [2006] Corner bifurcations in non-smoothly forced impact oscillators, Physica D 220, Chen, D., Wang, H. O. & Chen, G. [2001] Anti-control of Hopf bifurcation, IEEE Trans. Circuits Syst.-I 48, Chicone, C., Felts, K. & Mann, B. [2009] Observation and investigation of a non-monotonic period function for an impact oscillator, Commun. Nonlin. Sci. Numer. Simul. 14, di Bernardo, M., Budd, C., Champneys, A. R. & Kowalczyk, P. [2008] Piecewise-Smooth Dynamical Systems: Theory and Applications, 1st edition (Springer, London). Ding, W. C., Xie, J. H. & Sun, Q. G. [2004] Interaction of Hopf and period doubling bifurcations of a vibroimpact system, J. Sound Vib. 275, Divenyi,S.,Savi,M.A.,Weber,H.I.&Franca,L.F. P. [2008] Experimental investigation of an oscillator with discontinuous support considering different system aspects, Chaos Solit. Fract. 38, Foale, S. & Bishop, S. R. [1994] Bifurcations in impact oscillators: Theory and experiments, Nonlinearity and Chaos in Engineering Dynamics, ed. Thompson, J. M. T. (John Wiley & Sons Ltd., Chichester), pp Gan, C. B. & Lei, H. [2011] Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations, J. Sound Vib. 330, Kryzhevich, S. G. [2007] The symmetrization method and limit cycles of vibro-impact systems, Vestnik St. Petersburg Uni.: Math. 40, Li, Q. H., Chen, Y. M., Wei, Y. H. & Lu, Q. S. [2011] The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system, Int. J. Nonlin. Mech. 46, Luo, A. C. J. & Chen, L. [2005] Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts, Chaos Solit. Fract. 24,

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