The effects of non-stationary processes on chaotic and regular responses of the duffing oscillator

Moslehy, F.A.; Evan-Iwanowski, R. M.

doi:10.1016/0020-7462(91)90081-4

Cited by 16 publications

(10 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such Duffing-like systems contain the well-known saddle-node instabilities as illustrated in [3,14,15,25] among many others. This instability results in the familiar sudden jumps between stable amplitudes as system parameters change (e.g., excitation frequency).…”

Section: Resonance Shifting and The Jump Bifurcationmentioning

confidence: 99%

“…The effect of nonstationary influences on nonlinear systems includes predicting instabilities using transient dynamic effects [24,25], studying nonlinear resonant effects in rotating shafts [9,22], and the nonlinear response of a clamped plate under nonstationary frequency excitation [12]. Finally, extensive analytical and numerical investigation has been undertaken of nonstationary processes in general Duffing-type systems and their associated bifurcations [2,14] and in selfexcited and parametrically-excited systems [6,16]. The work in the present paper considers a relatively slow evolution rate, where the forcing frequency is a linear function of time; the evolution is slow, but nonnegligible over the time scales considered.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The nonstationary transition through resonance

Todd

Virgin

Gottwald³

1996

Nonlinear Dyn

View full text Add to dashboard Cite

This paper considers the resonant behavior of a mechanical oscillator during a linear frequency sweep. Both numerical and experimental results are presented. The experimental system consisting of a track in the shape of a potential energy surfaces has been used to highlight other types of nonlinear behavior and is here adapted so that the forcing frequency can be evolved continuously in time. The classic linear oscillator (with a parabolic potential well) is used as an introduction to illustrate basic features of the experiment and its response. Then, a track with a double well is used to assess nonstationary frequency effects on certain nonlinear characteristics, specifically amplitude jumps and flip bifurcations.

show abstract

Section: Resonance Shifting and The Jump Bifurcationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The nonstationary transition through resonance

Todd

Virgin

Gottwald³

1996

Nonlinear Dyn

View full text Add to dashboard Cite

show abstract

“…The work by Tran and Evan-Iwanowski [13] uncovers possible chaos generated by linear nonstationary inputs into the relaxation systems. The paper by Moslehy and Evan-Iwanowski [14] deals with the effects of nonstationary processes on the chaotic and regular stationary attractors in Duffing's oscillators. The paper by Evan-Iwanowski and Abhyankar [15] discusses the nonstationary panoramic transitions through chaos in the Duffing oscillators, which are characterized by the annihilation of the chaos and subsequent appearances of a variety of periodic symmetrical and asymmetrical responses.…”

Section: Introductionmentioning

confidence: 99%

Period doubling bifurcation problems in the softening Duffing oscillator with nonstationary excitation

Evan-Iwanowski

1994

Nonlinear Dyn

View full text Add to dashboard Cite

The Duffing oscillators are widely used to mathematically model a variety of engineering and physical systems. A computational analysis has been initiated to explore the effects of nonstationary excitations on the response of the softening Duffing oscillator in the region of the parameter space where the period doubling sequences occur. Significant differences between the stationary and nonstationary responses have been uncovered: (i) the stationary transitions from T to 2T, from 2T to 4T ... etc. branches at the stationary period doubling bifurcations are smooth, in nonstationary cases they exhibit jumps to the near stationary branches at the values of the control parameters greater than those for the stationary; this phenomenon is called penetration (delay or memory). The lengths of the penetrations is being compressed to zero with the increasing number of the iterations. (ii) The stationary and nonstationary responses eventually settle on different limit motions, the nonstationary has modulated components. (iii) The jumps appearing in the stationary bifurcation diagram at 2T from the upper to the lower branches of the (z, f) and (z, ~), i.e., (displacement-forcing amplitude) and (displacement-forcing frequency), diagrams have been replaced by continuous transition in the nonstationary diagram eliminating thus the discontinuity. Apart from these differences, some specific characteristic nonstationary responses have been observed not encountered in the stationary cases: (iv) the appearance of the 'window' in the nonstationary limit bifurcation diagrams. (v) The nonstationary limit motions located on the upper (lower) branches of the (z, f) or (z, ~) diagrams expanded rapidly to the lower (upper) branches. (vi) The stationary and nonstationary bifurcation diagrams are extremely sensitive to the initial conditions, manifested by the mirror reflections, called the flipflop phenomenon. (vii) The nonstationary limit motion has been characterized by a complex phase portrait, the appearance of the Cantor-like set of the limit motion bifurcation plot, and continuous spectral density. For the purpose of comparison, a stationary period doubling sequence T, 2T,..., 2'~T,... stationary limit motion, Xs~ which is known to be chaotic has been determined, A far reaching observation has been made in the process of this study: the determination of the nonstationary bifurcations, their branches and limit motions, has been independent of the calculations of the stationary ones, indicating, thus, the existence of an independent class of nonstationary (time varying) dynamics. the values of the varying bifurcations at which the NS responses jump from 2nT to 2'~+lT to the near ST branch. NS jumps occur after the ST transitions from 2'~T to 2'~+lT period doubling bifurcations. The difference between ST and NS bifurcation is called penetration, delay or memory. Xsx, ST period doubling converge to a ST limit motion or chaos. XNS, NS period doubling sequence converges to a NS limit motion or chaos. It is characterized by fractal geometry,...

show abstract

“…on the disturbance) and the system parameters. Moslehy and Evan-Iwanowski [8] considered excitations with varying amplitude and frequency acting on a Duffing oscillator and found that the faster the sweep rate, the earlier the nonstationary response departs from the stationary one.…”

Section: Introductionmentioning

confidence: 99%

Whirling of a forced cantilevered beam with static deflection. III: Passage through resonance

1993

View full text Add to dashboard Cite

Nonstationary excitations of slender, elastic, cantilevered beams with equal principal moments of inertia are considered. The excitation frequency is slowly increased or decreased through a resonance of the first mode at a constant rate. Three resonances are investigated: primary resonance, superharmonic resonance of order two and subharmonie resonance of order two. After application of Galerkin's method with three modes, the nonlinear, nonstationary response of the first mode of the beam is determined by two methods: integration of the modulation equations obtained from the method of multiple scales, and direct numerical integration of the temporal equations of motion. Time histories are presented and the effects of excitation amplitude, rate of acceleration or deceleration through resonance, damping and initial conditions of the disturbance on the maximum response are studied. The effect of a persistent random disturbance is also examined. Although the excitation acts in the vertical plane, whirling occurs if the beam is subjected to out-of-plane disturbances.

show abstract

The effects of non-stationary processes on chaotic and regular responses of the duffing oscillator

Cited by 16 publications

References 19 publications

The nonstationary transition through resonance

The nonstationary transition through resonance

Period doubling bifurcation problems in the softening Duffing oscillator with nonstationary excitation

Whirling of a forced cantilevered beam with static deflection. III: Passage through resonance

Contact Info

Product

Resources

About