skip to main content


Title: The effects of nonlinear damping on degenerate parametric amplification
Abstract This paper considers the dynamic response of a single degree of freedom system with nonlinear stiffness and nonlinear damping that is subjected to both resonant direct excitation and resonant parametric excitation, with a general phase between the two. This generalizes and expands on previous studies of nonlinear effects on parametric amplification, notably by including the effects of nonlinear damping, which is commonly observed in a large variety of systems, including micro- and nano-scale resonators. Using the method of averaging, a thorough parameter study is carried out that describes the effects of the amplitudes and relative phase of the two forms of excitation. The effects of nonlinear damping on the parametric gain are first derived. The transitions among various topological forms of the frequency response curves, which can include isolae, dual peaks, and loops, are determined, and bifurcation analyses in parameter spaces of interest are carried out. In general, these results provide a complete picture of the system response and allow one to select drive conditions of interest that avoid bistability while providing maximum amplitude gain, maximum phase sensitivity, or a flat resonant peak, in systems with nonlinear damping.  more » « less
Award ID(s):
1662619
NSF-PAR ID:
10296711
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Nonlinear Dynamics
Volume:
102
Issue:
4
ISSN:
0924-090X
Page Range / eLocation ID:
2433 to 2452
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. The chief objective of this paper is to explore energy transfer mechanism between the sub-systems that are coupled by a nonlinear elastic path. In the proposed model (via a minimal order, two degree of freedom system), both sub-systems are defined as damped harmonic oscillators with linear springs and dampers. The first sub-system is attached to the ground on one side but connected to the second sub-system on the other side. In addition, linear elastic and dissipative characteristics of both oscillators are assumed to be identical, and a harmonic force excitation is applied only on the mass element of second oscillator. The nonlinear spring (placed in between the two sub-systems) is assumed to exhibit cubic, hardening type nonlinearity. First, the governing equations of the two degree of freedom system with a nonlinear elastic path are obtained. Second, the nonlinear differential equations are solved with a semi-analytical (multi-term harmonic balance) method, and nonlinear frequency responses of the system are calculated for different path coupling cases. As such, the nonlinear path stiffness is gradually increased so that the stiffness ratio of nonlinear element to the linear element is 0.01, 0.05, 0.1, 0.5 and 1.0 while the absolute value of linear spring stiffness is kept intact. In all solutions, it is observed that the frequency response curves at the vicinity of resonant frequencies bend towards higher frequencies as expected due to the hardening effect. However, at moderate or higher levels of path coupling (say 0.1, 0.5 and 1.0), additional branches emerge in the frequency response curves but only at the first resonant frequency. This is due to higher displacement amplitudes at the first resonant frequency as compared to the second one. Even though the oscillators move in-phase around the first natural frequency, high amplitudes increase the contribution of the stored potential energy in the nonlinear spring to the total mechanical energy. The out-of-phase motion around the second natural frequency cannot significantly contribute due to very low motion amplitudes. Finally, the governing equations are numerically solved for the same levels of nonlinearity, and the motion responses of both sub-systems are calculated. Both in-phase and out-of-phase motion responses are successfully shown in numerical solutions, and phase portraits of the system are generated in order to illustrate its nonlinear dynamics. In conclusion, a better understanding of the effect of nonlinear elastic path on two damped harmonic oscillators is gained. 
    more » « less
  2. In this work we present a systematic review of novel and interesting behaviour we have observed in a simplified model of a MEMS oscillator. The model is third order and nonlinear, and we expressit as a single ODE for a displacement variable. We find that a single oscillator exhibits limitcycles whose amplitude is well approximated by perturbation methods. Two coupled identicaloscillators have in-phase and out-of-phase modes as well as more complicated motions.Bothof the simple modes are stable in some regions of the parameter space while the bifurcationstructure is quite complex in other regions. This structure is symmetric; the symmetry is brokenby the introduction of detuning between the two oscillators. Numerical integration of the fullsystem is used to check all bifurcation computations. Each individual oscillator is based on a MEMS structure which moves within a laser-driven interference pattern. As the structure vibrates, it changes the interference gap, causing the quantity of absorbed light to change, producing a feedback loop between the motion and the absorbed light and resulting in a limit cycle oscillation. A simplified model of this MEMS oscillator, omitting parametric feedback and structural damping, is investigated using Lindstedt's perturbation method. Conditions are derived on the parameters of the model for a limit cycle to exist. The original model of the MEMS oscillator consists of two equations: a second order ODE which describes the physical motion of a microbeam, and a first order ODE which describes the heat conduction due to the laser. Starting with these equations, we derive a single governing ODE which is of third order and which leads to the definition of a linear operator called the MEMS operator. The addition of nonlinear terms in the model is shown to produce limit cycle behavior. The differential equations of motion of the system of two coupled oscillators are numerically integrated for varying values of the coupling parameter. It is shown that the in-phase mode loses stability as the coupling parameter is reduced below a certain value, and is replaced by two new periodic motions which are born in a pitchfork bifurcation. Then as this parameter is further reduced, the form of the bifurcating periodic motions grows more complex, with yet additional bifurcations occurring. This sequence of bifurcations leads to a situation in which the only periodic motion is a stable out-of-phase mode. The complexity of the resulting sequence of bifurcations is illustrated through a series of diagrams based on numerical integration. 
    more » « less
  3. We have conducted three-dimensional (3D) 0–7.5 Hz physics-based wave propagation simulations to model the seismic response of the Long Valley Dam (LVD), which has formed Lake Crowley in Central California, to estimate peak ground motions and settlement of the dam expected during maximum credible earthquake (MCE) scenarios on the nearby Hilton Creek Fault (HCF). We calibrated the velocity structure, anelastic attenuation model, and the overall elastic properties of the dam via linear simulations of a Mw3.7 event as well as the Mw6.2 Chalfant Valley earthquake of 1986, constrained by observed ground motions on and nearby the LVD. The Statewide California Earthquake Center (SCEC) Community Velocity Model CVM-S4.26.M01 superimposed with a geotechnical layer using [Formula: see text] information tapered from the surface to a 700-m depth was used in the simulations. We found optimal fit of simulated and observed ground motions at the LVD using frequency-independent attenuation of [Formula: see text] ([Formula: see text] in m/s). Using the calibrated model, we simulated 3D nonlinear ground motions at the LVD for Mw6.6 rupture scenarios on the HCF using an Iwan-type, multi-yield-surface technique. We use a two-step method where the computationally expensive nonlinear calculations were carried out in a small domain with the plane wave excitation along the bottom boundary obtained from a full-domain 3D linear finite-fault simulation. Our nonlinear MCE simulation results show that peak ground velocities (PGVs) and peak ground accelerations (PGAs) as high as 72 cm/s and 0.55 g, respectively, can be expected at the crest of the LVD. Compared with linear ground motion simulation results, our results show that Iwan nonlinear damping reduces PGAs on the dam crest by up to a factor of 8 and increasingly depletes the high-frequency content of the waves toward the dam crest. We find horizontal relative displacements of the material inside the dam of up to [Formula: see text] and up to [Formula: see text] of vertical subsidence, equivalent to 1% of the dam height.

     
    more » « less
  4. Three dimensional dynamic soil-pile group interaction has been a subject of significant research interest over the past several decades, and remains an active and challenging topic in geotechnical engineering. A variety of dynamic excitation sources may potentially induce instabilities or even failures of pile groups. Employing modern experimental and numerical techniques, the dynamics of pile groups is examined in this study by integrated physical and computational simulations. In the physical phase, full- scale in-situ elastodynamic vibration tests were conducted on a single pile and a 2×2 pile group. Comprehensive site investigations were conducted for obtaining critical soil parameters for use in dynamic analyses. Broadband random excitation was applied to the pile cap and the response of the pile and soil were measured, with the results presented in multiple forms to reveal the dynamic characteristics of the pile-soil system. In the computational phase, the BEM code BEASSI was extended and modified to enable analysis of 3D dynamic pile group problems, and the new code was validated and verified by comparison to reference cases from the literature. A new theoretical formulation for analysis of multi-modal vibration of pile groups by accelerance functions is established using the method of sub-structuring. Various methods for interpreting the numerical results are presented and discussed. Case studies and further calibration of the BEM soil profiles are conducted to optimize the match between the theoretical and experimental accelerance functions. Parametric studies are performed to quantify the influence of the primary factors in the soil-pile system. It is shown that the new 3D disturbed zone continuum models can help improve the accuracy of dynamic soil-pile interaction analysis for pile groups in layered soils. This study therefore helps to advance the fundamental knowledge on dynamic soil-pile interaction by improving the accuracy of current computational models, and contributes additional physical tests to the experimental database in the literature. The specific impedance functions generated herein can be immediately used in practice, and the underlying general 3D disturbed-zone computational framework can readily be applied to other pile group problems of interest to researchers and practitioners. 
    more » « less
  5. Abstract

    Nonreciprocal transmission forms the basic operation mechanism of optical diodes and isolators and requires the tantalizing task of breaking the Lorentz reciprocity law. In this work, strong nonreciprocal transmission is demonstrated by using a compact nonlinear parity‐time (PT) symmetric system based on epsilon‐near‐zero (ENZ) materials photonically doped with gain and loss defects and separated by an ultrathin air gap. The nonlinear response of this scalable configuration is triggered at relatively low optical intensities due to the strong electric field confinement in the defects. The extreme asymmetric field distribution achieved upon excitation from opposite incident directions, combined with the enhanced nonlinear properties of the proposed system, results in a pronounced self‐induced nonreciprocal transmission. Cascade configurations with optimized geometrical dimensions are used to achieve self‐induced nonreciprocal transmission with a maximum contrast, ideal for the design of new all‐optical diodes. The presented robust nonreciprocal response occurs by operating at a frequency slightly shifted off the exceptional point but without breaking the PT‐symmetric phase, different compared to previous works. The findings of this work can have a plethora of applications, such as nonreciprocal ultrathin coatings for the protection of sources or other sensitive equipment from external pulsed signals, circulators, and isolators.

     
    more » « less