1
will belarge. Since thespecific shape of the nonlinearities will determine the rateat which different modes decay, a monotonic decrease willbe observed indimensionality until only the resonant modes willstill be active at small amplitudes before the next kick. Instead ofobserving the specific dynamics of the system, the dimen- sional complexity ofthe system is considered as a new dynamical variable. The time evolution ofthe dimension- al complexity parameter is compared with the original timeseries andalso with recurrence plots of the embedded time series. 4:05 9SA6. Vibration control of flexible structures via nonlinear coupling. Farid Golnaraghi (Dept. of Mech. Eng.,Univ. of Waterloo, Waterloo, OntarioN2L 3G1, Canada) In this paper, a rather unconventional active/passive vibration suppression technique fora cantilever beam based onsimple principles in nonlinear vibrations is proposed. Thecontroller is a sliding mass-spring-dashpot mechanism placed at the free end ofthe beam, introducing Coriolis, inertia, and centripetal nonlinearities into the system. These nonlinear terms may be used to eliminate the transient vibration of the beam whenthe natural frequency of the slider is twice the fundamental beam frequency (internal resonance). Once at 2:1 internal resonance, theoscillation energy istransferred fromthe beam to theslider and dissipated forsufficient- ly highslider damping. Numerical results show that thistechnique maybe used to increase the effective damping ratio ofthe structure from 0.5% to •6%. Thecontroller isparticularly successful in reducing large amplitudes ofoscillation to levels that may be handled byconventional methods. Forlower orzero controller damping, subharmonic andchaotic transient oscillations occur depending on theamplitude of the initial disturbance ofthe beam. Physical experiments are currently underway toverify the numerical and theoretical results. Contributed Papers 4:35 9SA7. Synchronization of chaotic systems. P. G. Vaidya, Rong He (Dept.ofMech. and Mater.Eng., Washington State Univ., Pullman, WA 99164), and M. J. Anderson (Univ. of Idaho, Moscow, ID 83843) Consider two identical chaotic systems that arestarted from virtually identical initial conditions. In a short time, they would be observed to diverge from oneanother. While it is true that, given ample passage of time, the systems will comewithin an arbitrarily close distance of one another, they would drift apart onceagain.In their fascinating paper, Pecora andCarroll have recently described a system of two Lorenzoscil- lators. They could be metaphorically described as a master and a slave system. The master system, undergoing chaos, drives a part of the slave system. Incredibly enough, the two systems remain in perfect synchroni- zation. In ourpaper, further analysis of this system ispresented. In addi- tion, another setof oscillators are proposed. These can perhaps be de- scribed assister-sister systems. Both influence oneanother, and control andmodification is possible with theaccess to either one. Some practical implications of these results are discussed. 4:50 9SA8. Routes to chaos in a geared system with backlash. AhmetKahraman a) andRaj Singh (Dept. of Mech. Eng., The Ohio State Univ., 206 W. 18th Av., Columbus, OH 43210-1107) A lumped parameter torsional-bending model of a spur gear pairwith backlash and time-invariant mesh stiffness is used to examine chaos. It can be shown that chaos typically exists in a lightly loaded and lightly damped gearpair whenexcited by the static transmission error. These results are qualitatively, but not quantitatively, similar to the studies re- ported on Duffing's oscillator. Whenradial clearances in rolling-element bearings areintroduced, a three degree of freedom nonlinear model isused to identify two differentroutesto chaos: (i) period-doubling route and (ii) quasiperiodic route. Predicted timehistories, phase planes, Poincar6 maps, andFourier spectra will be illustrated to examine these. Limited experimental data, as available in theliterature will becited in evidence. [Work supported byNASA Lewis Res. Ctr. ] a) Currently withGeneral Motors Res. Labs., Warren, MI 48090-9055. 5:05 9SA9. On the spectrum of random nonlinear vibration. Huw G. Davies and Qiang Liu (Dept. of Mech. Eng., Univ. of New Brunswick, Fredericton, NB E3B 5A3, Canada) Theresponse ofa nonlinear oscillator excited bywhite noise isconsid- ered. A truncated series of Hermitepolynomials isused asan approxima- tion to the probability density function. This series and the associated Fokker-Planck equation are used to generate twosets ofcoupled differen- tialequations fortime-dependent moments. Thefirst set is formoments of variables evaluated at the same time;the solution yields, for example, the nonstationary mean-square value. The second set isfor moments of vari- ables evaluated at two different times. This second set uses the solution of the firstsetasinitial conditions. A single-sided Fouriertransform of the second set yields coupled complex algebraic equations thatcan besolved numerically for the spectrum. Examples are shown of spectra for the Duffing oscillator showing an increase in effective resonance frequency and broadening ofthepeak as theexcitation level isincreased, and for the van der Pol oscillator showing an entrained limit cycle response at low excitation level thatdisappears as theexcitation level isincreased. [Work supported by NSERC, ] 5:20 9SA10. On the chaotic motion and acousticcharacteristicsof confined jet flows. Karo W. Ng (Naval Underwater Systems Ctr., Newport, RI 02841 ) and Nigel Lee (Princeton Univ., Princeton, NJ 08544) S195 J. Acoust. Soc.Am.Suppl. 1, Vol.88, Fall 1990 120thMeeting: Acoustical Society of America S195 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 77.53.187.129 On: Sun, 06 Apr 2014 09:45:31

Synchronization of chaotic systems

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will be large. Since the specific shape of the nonlinearities will determine the rate at which different modes decay, a monotonic decrease will be observed in dimensionality until only the resonant modes will still be active at small amplitudes before the next kick. Instead of observing the specific dynamics of the system, the dimen- sional complexity of the system is considered as a new dynamical variable. The time evolution of the dimension- al complexity parameter is compared with the original time series and also with recurrence plots of the embedded time series.

4:05

9SA6. Vibration control of flexible structures via nonlinear coupling. Farid Golnaraghi (Dept. of Mech. Eng., Univ. of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

In this paper, a rather unconventional active/passive vibration suppression technique for a cantilever beam based on simple principles in nonlinear vibrations is proposed. The controller is a sliding mass-spring-dashpot mechanism placed at the free end of the beam, introducing Coriolis, inertia, and centripetal nonlinearities into the system. These nonlinear terms may be used to eliminate the transient vibration of the beam when the natural frequency of the slider is twice the fundamental beam frequency (internal resonance). Once at 2:1 internal resonance, the oscillation energy is transferred from the beam to the slider and dissipated for sufficient- ly high slider damping. Numerical results show that this technique may be used to increase the effective damping ratio of the structure from 0.5% to •6%. The controller is particularly successful in reducing large amplitudes of oscillation to levels that may be handled by conventional methods. For lower or zero controller damping, subharmonic and chaotic transient oscillations occur depending on the amplitude of the initial disturbance of the beam. Physical experiments are currently underway to verify the numerical and theoretical results.

Contributed Papers

4:35

9SA7. Synchronization of chaotic systems. P. G. Vaidya, Rong He (Dept. of Mech. and Mater. Eng., Washington State Univ., Pullman, WA 99164), and M. J. Anderson (Univ. of Idaho, Moscow, ID 83843)

Consider two identical chaotic systems that are started from virtually identical initial conditions. In a short time, they would be observed to diverge from one another. While it is true that, given ample passage of time, the systems will come within an arbitrarily close distance of one another, they would drift apart once again. In their fascinating paper, Pecora and Carroll have recently described a system of two Lorenz oscil- lators. They could be metaphorically described as a master and a slave system. The master system, undergoing chaos, drives a part of the slave system. Incredibly enough, the two systems remain in perfect synchroni- zation. In our paper, further analysis of this system is presented. In addi- tion, another set of oscillators are proposed. These can perhaps be de- scribed as sister-sister systems. Both influence one another, and control and modification is possible with the access to either one. Some practical implications of these results are discussed.

4:50

9SA8. Routes to chaos in a geared system with backlash. Ahmet Kahraman a) and Raj Singh (Dept. of Mech. Eng., The Ohio State Univ., 206 W. 18th Av., Columbus, OH 43210-1107)

A lumped parameter torsional-bending model of a spur gear pair with backlash and time-invariant mesh stiffness is used to examine chaos. It

can be shown that chaos typically exists in a lightly loaded and lightly damped gear pair when excited by the static transmission error. These results are qualitatively, but not quantitatively, similar to the studies re- ported on Duffing's oscillator. When radial clearances in rolling-element bearings are introduced, a three degree of freedom nonlinear model is used to identify two different routes to chaos: (i) period-doubling route and (ii) quasiperiodic route. Predicted time histories, phase planes, Poincar6

maps, and Fourier spectra will be illustrated to examine these. Limited experimental data, as available in the literature will be cited in evidence. [Work supported by NASA Lewis Res. Ctr. ] a) Currently with General Motors Res. Labs., Warren, MI 48090-9055.

5:05

9SA9. On the spectrum of random nonlinear vibration. Huw G. Davies and Qiang Liu (Dept. of Mech. Eng., Univ. of New Brunswick, Fredericton, NB E3B 5A3, Canada)

The response of a nonlinear oscillator excited by white noise is consid- ered. A truncated series of Hermite polynomials is used as an approxima- tion to the probability density function. This series and the associated Fokker-Planck equation are used to generate two sets of coupled differen- tial equations for time-dependent moments. The first set is for moments of variables evaluated at the same time; the solution yields, for example, the nonstationary mean-square value. The second set is for moments of vari- ables evaluated at two different times. This second set uses the solution of

the first set as initial conditions. A single-sided Fourier transform of the second set yields coupled complex algebraic equations that can be solved numerically for the spectrum. Examples are shown of spectra for the Duffing oscillator showing an increase in effective resonance frequency and broadening of the peak as the excitation level is increased, and for the van der Pol oscillator showing an entrained limit cycle response at low excitation level that disappears as the excitation level is increased. [ Work supported by NSERC, ]

5:20

9SA10. On the chaotic motion and acoustic characteristics of confined jet flows. Karo W. Ng (Naval Underwater Systems Ctr., Newport, RI 02841 ) and Nigel Lee (Princeton Univ., Princeton, NJ 08544)

S195 J. Acoust. Soc. Am. Suppl. 1, Vol. 88, Fall 1990 120th Meeting: Acoustical Society of America S195

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 77.53.187.129 On: Sun, 06 Apr 2014 09:45:31