Stability Analysis of a Moored Vessel

2004 ◽  
Vol 126 (2) ◽  
pp. 164-174
Author(s):  
A. Umar ◽  
S. Ahmad ◽  
T. K. Datta
Keyword(s):  
Stability Analysis ◽  
Harmonic Balance ◽  
Variational Approach ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Periodic Wave ◽  
Mooring System ◽  
Wave Excitation ◽  
The Stability ◽  
Slack Mooring

A procedure for the stability analysis of a slack mooring system is presented for periodic wave excitation by finding its approximate response using a two term harmonic balance method (HBM). The conditions for determining the local and global stability of the approximate solutions are established using Hill’s variational approach and Floquet’s theory. A number of instability phenomena are identified for the mooring system for certain frequencies of excitations which fall outside the range of frequencies obtained from the analytically derived stability boundaries. The instability phenomena include symmetry breaking bifurcation, subharmonics, 3T and 5T solutions. Even chaotic motion is exhibited under certain cases.

Parametric Stability Analysis of Nonlinear Rotating Blades

2011 ◽  
Author(s):  
Fengxia Wang
Keyword(s):  
Stability Analysis ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Rotating Blade ◽  
Geometric Stiffening ◽  
Established Fact ◽  
The Stability ◽  
Stiffening Effect ◽  
Generalized Harmonic Balance Method

The role of the “geometric stiffening” nonlinearities played in the stability analysis of a rotating beam is investigated. It is a well established fact that nonlinear theory must be employed to capture geometric stiffening effect, which has been extensively investigated. In this work, two models are built for a rotating blade with periodically perturbed rotation rate, one is the “effective load” linear model and the other is “geometric stiffening” nonlinear model. Both of these two models are discretisized via Galerkin’s method and a set of parametric excited gyroscopic equations are obtained. The dynamic stability of these two models are studied and compared by the generalized harmonic balance method.

Multiple Harmonic Balance Method for Aperiodic Vibration of a Piecewise-Linear System

1998 ◽  
Vol 120 (1) ◽  
pp. 181-187 ◽  
Author(s):  
Y. B. Kim
Keyword(s):  
Linear System ◽  
Harmonic Balance ◽  
Piecewise Linear ◽  
Harmonic Balance Method ◽  
Steady State Solution ◽  
Balance Method ◽  
Piecewise Linear System ◽  
Higher Dimensional ◽  
Computational Procedures ◽  
The Stability

A multiple harmonic balance method is presented in this paper for obtaining the aperiodic steady-state solution of a piecewise-linear system. As the method utilizes general and systematic computational procedures, it can be applied to analyze the multi-tone or combination-tone responses for the higher dimensional nonlinear systems such as rotors. Moreover, it is capable of informing the stability of the obtained solution using Floquet theory. To demonstrate the systematic approach of the new method, the almost periodic forced vibration of an articulated loading platform (ALP) with a piecewise-linear stiffness is computed as an example.

scholarly journals The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Y. H. Qian ◽  
J. L. Pan ◽  
S. P. Chen ◽  
M. H. Yao
Keyword(s):  
Exact Solutions ◽  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Harmonic Balance Method ◽  
Time History ◽  
Approximate Solutions ◽  
Balance Method ◽  
Lumped Mass ◽  
Strongly Nonlinear

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

The Residue Harmonic Balance Method for Duffing-Van Der Pol Oscillator with Fractional Derivative

2013 ◽  
Vol 774-776 ◽  
pp. 103-106
Author(s):  
Xin Xue ◽  
Lian Zhong Li ◽  
Dan Sun
Keyword(s):  
Fractional Derivative ◽  
Harmonic Balance ◽  
Numerical Solutions ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Solution Procedure ◽  
Balance Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Fractional Orders

Duffing-van der Pol oscillator with fractional derivative was constructed in this paper. The solution procedure was proposed with the residue harmonic balance method. The effect of different fractional orders on resonance responses of the system in steady state were analyzed for an example without parameters. The approximate solutions were contrasted with numerical solutions. The results show that the residue harmonic balance method to Duffing-van der Pol differential equation with fractional derivative is very valid.

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Periodic Motions ◽  
Conditions Of Stability ◽  
Generalized Harmonic Balance Method

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

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 Harmonic Balance Solution Of Coupled Nonlinear Non-Conservative Differential Equation

2013 ◽  
Vol 32 ◽  
pp. 1-14
Author(s):  
M Saifur Rahman ◽  
M Majedur Rahman ◽  
M Sajedur Rahaman ◽  
M Shamsul Alam
Keyword(s):  
Differential Equation ◽  
Numerical Solution ◽  
Limit Cycle ◽  
Nonlinear Differential Equation ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Balance Solution ◽  
Good Agreement

A modified harmonic balance method is employed to determine the second approximate solutions to a coupled nonlinear differential equation near the limit cycle. The solution shows a good agreement with the numerical solution. DOI: http://dx.doi.org/10.3329/ganit.v32i0.13640 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 1 – 14

scholarly journals A Harmonic-Based Method for Computing the Stability of Periodic Oscillations of Non-Linear Structural Systems

2010 ◽  
Author(s):  
O. Thomas ◽  
A. Lazarus ◽  
C. Touze´
Keyword(s):  
Numerical Method ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Computation Time ◽  
Fourier Series Expansion ◽  
Simple Strategy ◽  
The Stability ◽  
Linear Periodic System ◽  
Continuation Technique

In this paper, we present a validation on a practical example of a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on Floquet theory and Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the Harmonic Balance Method and Asymptotic Numerical Method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. This procedure is validated by considering an externally forced string and computing the complete bifurcation diagram with the stability of the periodic solutions. The particular coupled regimes are exhibited and found in excellent agreement with results of the literature, allowing a method validation.

Parametric Stability of a Nonlinear Rotating Blade

2013 ◽  
Author(s):  
Fengxia Wang ◽  
Albert C. J. Luo
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Parametric Stability ◽  
Rotating Blade ◽  
Geometric Term ◽  
Stability And Bifurcations ◽  
The Stability ◽  
The Galerkin Method ◽  
Gyroscopic Systems ◽  
Generalized Harmonic Balance Method

The stability of period-1 motions of a rotating blade with geometric nonlinearity is studied. The roles of cubic stiffening geometric term are considered in the study of nonlinear periodic motions and its stability and bifurcations of a rotating blade. The nonlinear model of a rotating blade is reduced to the ordinary differential equations through the Galerkin method, and the gyroscopic systems with parametric excitations are obtained. The generalized harmonic balance method is employed to determine the period-1 solutions and the corresponding stability and bifurcations.

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