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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
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