The effects of nonlinear damping on degenerate parametric amplification

Li, Donghao; Shaw, Steven W.

doi:10.1007/s11071-020-06090-8

Cited by 25 publications

(12 citation statements)

References 70 publications

(121 reference statements)

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“…As recently shown in Ref. [20], Eqs. ( 22)-( 23) can generate intricate amplitude response curves that differ drastically from the standard directly driven Duffing resonator and include isolae, dual peaks, loops, and flat resonant peaks.…”

Section: The Case Of ω F2 = 2ω F1supporting

confidence: 61%

“…However, we are interested here in a relatively simple response curve, which is amplified by the induced parametric excitation from the secondary resonator. To this end, we set θ = −π/4 [20] and use the following set of system parameters: ω 1 = 1, ω 2 = 10, Γ 1 = 10 −3 , Γ 2 = 1, F 1 = 10 −3 , F 2 = 1, G 1 = 0, G 2 = 5, γ = 0.13, which yields a drastic reduction in the Duffing nonlinearityγ/γ = 2.57 × 10 −2 (97.43%), a cubic nonlinear damping with a coefficient of α 2 Γ 2 = 2.7 × 10 −3 , and a parametric drive level of k 1 G 2 F 2 = 0.05. The numerically validated modified response curve is shown in Fig.…”

Section: The Case Of ω F2 = 2ω F1mentioning

confidence: 99%

“…The importance of signal amplification stems from the need to operate above the noise-floor of the device with a large signal-to-noise ratio in order to use these tiny resonators for sensitive detection of displacement [1], mass [2], force [3], torque [4], and charge [5]. Due to its practical significance, signal amplification has received a considerable amount of attention [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Diverse methods have been devised for signal amplification including electrical means, such as lock-in amplifiers [21] and phase-locked loops [22], where the backaction from the inherently noisy electrical amplifier needs to be treated, and mechanical means, where the motion of the resonator is mechanically preamplified by a large factor before being transduced into an electrical signal.…”

Section: Introductionmentioning

confidence: 99%

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Amplifying the response of a driven resonator via nonlinear interaction with a secondary resonator

Rosenberg

Shoshani

2021

Nonlinear Dyn

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We study the dynamics of a resonantly driven nonlinear resonator (primary) that is nonlinearly coupled to a non-resonantly driven linear resonator (secondary) with a relatively short decay time. Due to its short relaxation time, the secondary resonator adiabatically tracks the primary resonator and modifies its response. Our model, which is motivated by experimental studies on the interaction between nano-and microresonators, is relatively simple and can be analyzed analytically and numerically to show that the driven response of the primary resonator can be enhanced significantly due to the interaction with the secondary resonator. Such an arrangement may pave the way for systematic control of driven responses and signal amplification in engineering applications involving nano-and micro-electro-mechanical-systems.

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Section: The Case Of ω F2 = 2ω F1supporting

confidence: 61%

Section: The Case Of ω F2 = 2ω F1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Amplifying the response of a driven resonator via nonlinear interaction with a secondary resonator

Rosenberg

Shoshani

2021

Nonlinear Dyn

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“…In [13], previous studies of the effects of nonlinear damping on the parametric gain are generalized and expanded.…”

Section: Introductionmentioning

confidence: 99%

Modeling the Dynamics of a Gyroscopic Rigid Rotor with Linear and Nonlinear Damping and Nonlinear Stiffness of the Elastic Support

Iskakov¹,

Bissembayev

Jamalov

et al. 2021

Machines

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This study analytically and numerically modeled the dynamics of a gyroscopic rigid rotor with linear and nonlinear cubic damping and nonlinear cubic stiffness of an elastic support. It has been shown that (i) joint linear and nonlinear cubic damping significantly suppresses the vibration amplitude (including the maximum) in the resonant velocity region and beyond it, and (ii) joint linear and nonlinear cubic damping more effectively affects the boundaries of the bistability region by its narrowing than linear damping. A methodology is proposed for determining and identifying the coefficients of nonlinear stiffness, linear damping, and nonlinear cubic damping of the support material, where jump-like effects are eliminated. Damping also affects the stability of motion; if linear damping shifts the left boundary of the instability region towards large amplitudes and speeds of rotation of the shaft, then nonlinear cubic damping can completely eliminate it. The varying amplitude (VAM) method is used to determine the nature of the system response, supplemented with the concept of “slow” time, which allows us to investigate and analyze the effect of nonlinear cubic damping and nonlinear rigidity of cubic order on the frequency response at a nonstationary resonant transition.

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“…Introduction. Complex nonlinear phenomena that occur in a nonlinear driven oscillator are the interest of worldwide studies over the past decades [8,9,12,11,13,1,5,2,17,16,4,15,10,14,7,3,6]. A paradigmatic family of systems are assumed in the form of the second order nonautonomous differential equations ẍ + µ ẋ + dV (x)/dx + f (x, t) = 0, (1) where x denotes a displacement from the equilibrium position, dots stand for differentiating with respect to time t, µ ẋ is a weak damping term, with parameter µ denoting the damping intensity, f (x, t) represents periodic function of time with the period T = 2π/ω, and V (x) is the potential energy approximated by a finite Taylor series.…”

mentioning

confidence: 99%

The damping term makes the Smale-horseshoe heteroclinic chaotic motion easier

Sun¹,

Cao²

2022

DCDS-B

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<p style='text-indent:20px;'>The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smale-horseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smale-horseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasi-periodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasi-periodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.</p>

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The effects of nonlinear damping on degenerate parametric amplification

Cited by 25 publications

References 70 publications

Amplifying the response of a driven resonator via nonlinear interaction with a secondary resonator

Amplifying the response of a driven resonator via nonlinear interaction with a secondary resonator

Modeling the Dynamics of a Gyroscopic Rigid Rotor with Linear and Nonlinear Damping and Nonlinear Stiffness of the Elastic Support

The damping term makes the Smale-horseshoe heteroclinic chaotic motion easier

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