On the Internal Resonance in a Nonlinear Two-Degree-of-Freedom System : Forced Vibrations near the Higher Resonance Point When the Natural Frequencies Are in the Ratio 1:2

Yamamoto, Toshio; Yasuda, Kimihiko; Nagasaka, Imao

doi:10.1299/jsme1958.20.1093

Cited by 11 publications

(5 citation statements)

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“…The general solutions of equations (6)- (8) can be expressed in the form u,,,, = A,,(TI) exp(i% To) + cc (12) for n = 1, 2, and 3. where cc denotes the complex conjugate of the preceding terms, and the A,, are to be determined through the elimination of secular and small-divisor terms from the next-order equations. In this paper, we analyze the case w 3 = 2~2, w 2 = 2w~.…”

Section: Methods Of Solution: Multiple Scalesmentioning

confidence: 99%

“…Substituting equations (51) into equations (12) and using equations (16) and (33), we find that to the first approximation OL 7 where the Pi and q, are given by equations (52)-(57). The stability of a particular fixed point with respect to perturbations proportional to exp( )t T~) depends on the real parts of the roots of the characteristic equation…”

Section: Stability Of the Fixed Pointsmentioning

confidence: 99%

“…Later, Yamamoto and co-workers [11,12] used the method of harmonic balance and analog-computer simulations to investigate the forced responses of systems with quadratic and cubic nonlinearities to harmonic excitations when one natural frequency is twice another. They observed amplitude-and phase-modulated steady-state motions in their analog-computer simulations when 1)~ o) 2 and ft ~ o)~.…”

Section: Introductionmentioning

confidence: 99%

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Three-mode interactions in harmonically excited systems with quadratic nonlinearities

1992

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An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances w 3 ~ 2w 2 and w 2 ~ 2~o~ to a harmonic excitation of the third mode, where the ~o,,, are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitude F of the excitation as a control parameter. As the excitation amplitude F is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. As F is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. As F is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.

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Section: Methods Of Solution: Multiple Scalesmentioning

confidence: 99%

Section: Stability Of the Fixed Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Three-mode interactions in harmonically excited systems with quadratic nonlinearities

1992

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“…These cases are (1, m, n, q) for m = 2, 4, n = 1, 2, q = 1, 3, (1, 4, n, 1) for n = 3, 4, and (1, m, 4, 3) for m = 2, 4, all with r = 1 2 or r = 3 2 . Both oscillators display quasiperiodic motion, resembling what has been described for similar systems in various ways including as "modulated", "almost periodic", and an "exchange of energy" [1,3,5,6,[22][23][24][25][26][27][28][29][30]. A numerical Fourier analysis reveals the appearance of second frequencies close to the natural frequencies of each oscillator, leading to beat frequencies that scale as f B ∼ .…”

Section: Resultsmentioning

confidence: 74%

The saturation bifurcation in coupled oscillators

2018

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We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent, unforced subsystem. We derive and numerically verify analytical predictions for the existence and behavior of such saturated states for a class of oscillator pairs. We also examine related phenomena, including zero-frequency response to periodic forcing, Hopf bifurcations to quasiperiodicity, and bifurcations to periodic behavior with multiple frequencies.

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“…This means that the excited second mode takes a small amount of the input energy and spills over the rest of the input energy into the first mode, which is indirectly excited through the internal resonance. Yamamoto et al [12] demonstrated amplitude-and phase-modulated motions and Hatwal et al…”

Section: Introductionmentioning

confidence: 99%

Nonlinear random coupled motions of structural elements with quadratic nonlinearities

Serhan¹,

Nayfeh

1991

Nonlinear Dyn

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The second-order closure method is used to analyze the nonlinear response of two-degree-of-freedom systems with quadratic nonlinearities. The excitation is assumed to be the sum of a deterministic harmonic component and a random component. The case of primary resonance of the second mode in the presence of a two-to-one internal (autoparametric) resonance is investigated. The method of multiple scales is used to obtain four first-order ordinarydifferential equations that describe the modulation of the amplitudes and phases of the two modes. Applying the second-order closure method to the modulation equations, we determine the stationary mean and mean-square responses. For the case of a narrow-band random excitation, the results show that the presence of the nonlinearity causes multi-valued mean-square responses. The multi-valuedness is responsible for a jump phenomenon. Contrary to the results of the linear analysis, the nonlinear analysis reveals that the directly excited second mode takes a small amount of the input energy (saturates) and spills over the rest of the input energy into the first mode, which is indirectly excited through the autoparametric resonance.

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On the Internal Resonance in a Nonlinear Two-Degree-of-Freedom System : Forced Vibrations near the Higher Resonance Point When the Natural Frequencies Are in the Ratio 1:2

Cited by 11 publications

References 0 publications

Three-mode interactions in harmonically excited systems with quadratic nonlinearities

Three-mode interactions in harmonically excited systems with quadratic nonlinearities

The saturation bifurcation in coupled oscillators

Nonlinear random coupled motions of structural elements with quadratic nonlinearities

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