Revealing Nonlinear Dynamics Phenomena in Mooring Due to Slowly-Varying Drift

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Michael M. Bernitsas ◽  
João Paulo J. Matsuura
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Resonance Phenomenon ◽  
Nonlinear Phenomena ◽  
Mooring System ◽  
Stable Limit ◽  
Dynamic Interactions ◽  
Wave Drift

The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drifts and mooring systems. We were able to reveal new phenomena, based on the design methodology developed at the University of Michigan for autonomous mooring systems, and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed, based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency, estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or nonperiodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes two to three orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.

Revealing Nonlinear Dynamics Phenomena in Mooring Due to Slowly-Varying Drift

2005 ◽  
Author(s):  
Michael M. Bernitsas ◽  
Joa˜o Paulo J. Matsuura
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Resonance Phenomenon ◽  
Nonlinear Phenomena ◽  
Mooring System ◽  
Stable Limit ◽  
Dynamic Interactions ◽  
Wave Drift

The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drift and mooring systems. We were able to reveal new phenomena based on the design methodology developed at the University of Michigan for autonomous mooring systems and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or non-periodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes 2–3 orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.

scholarly journals Phase-Coupled Oscillations in the Brain: Nonlinear Phenomena in Cellular Signalling

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Vikas Rai ◽  
Sreenivasan Rajamoni Nadar ◽  
Riaz A. Khan
Keyword(s):  
Limit Cycles ◽  
Initial Conditions ◽  
Phase Relationship ◽  
Oscillatory System ◽  
Calcium Currents ◽  
Nonlinear Phenomena ◽  
Inhibitory Interneurons ◽  
Stable Limit ◽  
Principal Cells ◽  
Coupled Oscillations

We report the existence of phase-coupled oscillations in a model neural system. The model consists of a group of excitatory principal cells in interaction with local inhibitory interneurons. The voltages across the membranes of excitatory cells are governed primarily by calcium and potassium ion conductivities. The number of potassium channels open at any given instant changes in accordance with a deterministic law. The time scale of this change is set by a constant which depends on midpoint potentials at which potassium and calcium currents are half-activated. The growth of mean membrane potential of excitatory principal cells is controlled by that of the inhibitory interneurons. Nonlinear oscillatory system associated with these limit cycles starting from two different initial conditions maintain a definite phase relationship. The phase-coupled oscillations in electrical activity of the neuronal cells carry together amplitude, phase, and time information for cellular signaling. This mechanism supports an energy efficient way of information processing in the central nervous system. The information content is encoded as persistent periodic oscillations represented by stable limit cycles in the phase space.

Analytical Solution for the Nonlinear Dynamics of Planetary Gears

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Cheon-Jae Bahk ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Response ◽  
Harmonic Balance ◽  
Gear Tooth ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Mesh Stiffness ◽  
Planetary Gears ◽  
Parallel Finite Element ◽  
The Impact

Planetary gears are parametrically excited by the time-varying mesh stiffness that fluctuates as the number of gear tooth pairs in contact changes during gear rotation. At resonance, the resulting vibration causes tooth separation leading to nonlinear effects such as jump phenomena and subharmonic resonance. This work examines the nonlinear dynamics of planetary gears by numerical and analytical methods over the meaningful mesh frequency ranges. Concise, closed-form approximations for the dynamic response are obtained by perturbation analysis. The analytical solutions give insight into the nonlinear dynamics and the impact of system parameters on dynamic response. Correlation between the amplitude of response and external torque demonstrates that tooth separation occurs even under large torque. The harmonic balance method with arclength continuation confirms the perturbation solutions. The accuracy of the analytical and harmonic balance solutions is evaluated by parallel finite element and numerical integration simulations.

scholarly journals Nonlinear Dynamic Behaviour Analysis of a Clutch System with Uncertainties Using Polynomial Chaos and the Constrained Harmonic Balance Method

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
M.-H. Trinh ◽  
S. Berger ◽  
E. Aubry
Keyword(s):  
Limit Cycles ◽  
Nonlinear Dynamic ◽  
Harmonic Balance ◽  
Dynamic Behaviour ◽  
Polynomial Chaos ◽  
Equilibrium Points ◽  
Harmonic Balance Method ◽  
Unstable Equilibrium ◽  
Balance Method ◽  
Clutch System

The study of the nonlinear dynamic behaviour of friction systems in general and of clutch systems in particular remains an open problem. Noise and vibrations induced by friction in the sliding phase of a clutch are very sensitive to design parameters. The latter have significant dispersions. In the study of the system stability, the problem is not only to know if the parameter values lead to the appearance of unstable equilibrium points; the real challenge lies in estimating the vibration levels when such unstable equilibrium points occur. This estimation is analyzed using the limit cycles. This article aims to study the ability of robust approaches based on developments in nonintrusive generalized polynomial chaos and a constrained harmonic balance method to estimate the vibration levels through the limit cycles of a clutch system in the presence of uncertainty. The purpose is to provide a low-cost, high precision approach, compared to the classic Monte Carlo method.

Nonlinear Dynamics of the Rod-Fastened Jeffcott Rotor

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Qi Yuan ◽  
Jin Gao ◽  
Pu Li
Keyword(s):  
Nonlinear Dynamics ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Operating Parameters ◽  
Flexural Stiffness ◽  
The Finite Element Method ◽  
Jeffcott Rotor ◽  
Unbalance Force ◽  
Contact Surfaces ◽  
Tie Rods

Rod-fastened rotors are composed of some disks clamped together by a central tie rod or several tie rods distributed along the circumference. Due to the nonlinear flexural stiffness of the contact interfaces in disks, especially when the contact surfaces are partially separated, the dynamics of the rod-fastened rotors are potentially different from that of the solid rotors. In this paper, the nonlinear flexural stiffness of a rod-fastened Jeffcott rotor is calculated by the finite element method (FEM). Then the harmonic balance method is adopted to analyze the dynamics of the rotor. The flexural stiffness of a rod-fastened Jeffcott rotor dramatically decreased with the increase of the dimensionless load γ1 from 1 to 2.5. Thus, the dynamics of the rotor were nonlinear when it was subjected to a large unbalance force. The response of the rotating rotor contains a predominantly forward 1X component or both forward 1X component and backward 1X components. However, the rotor may settle in a state depending upon both the operating parameters and its history.

A Harmonic Balance Approximation of Dynamic Snap-Through Boundaries in a Single-Degree-of-Freedom Structure

2013 ◽  
Author(s):  
Richard Wiebe ◽  
Lawrence N. Virgin
Keyword(s):  
Large Amplitude ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Nonlinear Phenomena ◽  
Amplitude Response ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Analytical Approximations

Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable. Structures with underlying snap-through static behavior may exhibit highly nonlinear and unpredictable oscillations. Such systems rarely lend themselves to investigation by analytical means. This is not surprising as nonlinear phenomena such as chaos run counter to the predictability of an analytical closed form solution. However, many unexpected analytical approximations of global stability may be obtained for simple systems using the harmonic balance method. In this paper a simple single-degree-of-freedom arch is studied using the harmonic balance method. The equations developed with the harmonic balance approach are then solved using an arc-length method and an approximate snap-through boundary in forcing parameter space is obtained. The method is shown to exhibit excellent agreement with numerical results. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many applications in both civil engineering, where arches are a canonical structural element, and mechanical/aerospace engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.

Nonlinear Dynamics in Drilling and Boring Operations Assisted by Low Frequency Vibration

2007 ◽  
Author(s):  
George F. Moraru
Keyword(s):  
Nonlinear Dynamics ◽  
Harmonic Balance ◽  
Low Frequency ◽  
Harmonic Balance Method ◽  
Nonlinear Behavior ◽  
Drilling Process ◽  
Process Damping ◽  
Regenerative Effect ◽  
Low Frequency Vibration ◽  
Machining System

The nonlinear dynamics of the drilling process assisted by self-excited axial vibrations is analyzed. Models are developed and discussed, including regenerative effect and various phenomena contributing to the process damping in drilling and boring operations. Stability and bifurcation analysis, using several assumptions on the damping in the cutting process, are carried out using linear analysis tools or simulations. A simple predictive model based on a harmonic balance method is presented. Behavior charts are constructed using simulations. Hypothesis on the nature of the process damping and nonlinear behavior of the machining system are proved by experiments. A possible application to the gundrilling boring process in aircraft drilling and assembly process is presented.

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