Generalized Harmonic Analysis of Nonlinear Ship Roll Dynamics

1996 ◽  
Vol 40 (04) ◽  
pp. 316-325
Author(s):  
J. C. Peyton Jones ◽  
I. Cankaya
Keyword(s):  
Harmonic Analysis ◽  
Harmonic Balance ◽  
Initial Conditions ◽  
Balance Equations ◽  
Validation Process ◽  
Algebraic Expressions ◽  
Resonant Modes ◽  
Ship Roll ◽  
Righting Moment ◽  
Simulation Time

Algebraic expressions are presented which enable the harmonic balance equations to be written down directly in terms of the coefficients of a general nonlinear ship roll equation, without restriction on the number of harmonics considered. The rolling response is then readily computed, as illustrated by an investigation of the resonant modes of ships with angle-dependent cubic damping, or quintic terms in the righting moment. Details are also given of the validation process, showing how simulation time can be reduced by an appropriate choice of initial conditions.

A Note on the Computation of Maximum Roll Amplitudes in Regular Beam Seas

1997 ◽  
Vol 41 (03) ◽  
pp. 224-229
Author(s):  
J. C. Peyton Jones ◽  
I. Qankaya
Keyword(s):  
Harmonic Balance ◽  
Wide Frequency Range ◽  
Phase Constraint ◽  
Frequency Range ◽  
Balance Equations ◽  
Efficient Tool ◽  
Beam Seas ◽  
At Resonance ◽  
Ship Roll ◽  
Cubic Stiffness

Generalized methods for deriving the harmonic balance equations are illustrated using a ship roll model with angle dependent cubic damping, and cubic stiffness terms. The balance equations are then solved subject to a phase constraint which identifies the resonant solution only. The amplitude of the response at resonance, and the frequency at which resonance occurs, can therefore be obtained without needing to compute the response over a wide frequency range. This provides an efficient tool for investigating the dependence of the resonant response on the level of input excitation, with results which agree well with more time-consuming simulation methods.

scholarly journals ABOUT OF THE DYNAMICS OF FORCED OSCILLATIONS IN NONLINEAR SYSTEMS

2019 ◽  
Author(s):  
A. N. Qurbanov ◽  
I. Z. Sardarova
Keyword(s):  
Nonlinear System ◽  
Harmonic Balance ◽  
Initial Conditions ◽  
Forced Oscillations ◽  
Dynamic Mode ◽  
Vibration Exciter ◽  
Balance Equations ◽  
Mechanical Part ◽  
Cyclic Mode ◽  
Electromagnetic Vibration

The article discusses the issues of determining the effectiveness indicators of an electromagnetic vibration exciter in a dynamic mode of operation. It is established that this system is whole system of electrical and mechanical circuits. In this case, the mechanical part operates in the mode of forced vibrations. The oscillation parameters of the system, such as amplitude, frequency, and phase, largely depend on the parameters of the system load. For the analysis of this system, differential equations describing an electromagnetic vibration exciter have been compiled. For this purpose, the dependence L(x) of the inductance on the displacement is used. The dynamic modes of one of the ways of asynchronous excitation of an electromagnetic vibration exciter are investigated. The accuracy analysis and the evaluation of the results were performed by the Fisher criterion for the regression model. To analysis of transients in the electromagnetic vibration exciter, were used the software packages WinFact and MatLab to simulate and optimize dynamic systems. It is established that the system, depending on the initial conditions in the simulation, goes into one of two very different modes. In this case, the initial zero conditions switch the system into a “cyclic” mode, and in other, non-zero conditions, the system goes into an approximate cyclic mode, characterized by a higher speed of movement of the anchor. The parameters of the steady state cyclic movement are determined by the method of harmonic balance. The obtained results allow us to describe autoparametric oscillations of the electric equivalent circuit. It has been established that the compilation of harmonic balance equations corresponding to a linear system helps simplify the solution of the task of determining the dynamics of forced oscillations. The expressions for determining the tractive force and the current flowing through the circuit are obtained, the wavelet spectra of vibration are constructed using the MatLab software package. As a result, for the mechanical part of a nonlinear system, in fact, it is necessary to solve only harmonic balance equation. The results show that this theoretical model allows a more qualitative and accurate assessment of the observed phenomenon. Based on this, the asymptotic conditions for solving the harmonic balance equations of a nonlinear system are determined. The expressions for the electromagnetic force acting on the anchor are obtained, the conditions for the harmonic balance of the mechanical part of the system are determined. The expressions obtained allow us to construct the amplitude-frequency characteristics of the electromagnetic vibration exciter. In conclusion, not only qualitative, but also quantitative estimates of the observed phenomena were obtained. It has been established that mechanical oscillations of a nonlinear system are insensitive to changes in the supply network and practically have a large amplitude with a constant frequency.

THE NEW ALGORITHM OF SOLVING HARMONIC BALANCE EQUATIONS

2019 ◽  
Author(s):  
Vladimir Lantsov ◽  
A. Papulina
Keyword(s):  
Harmonic Balance ◽  
Model Order Reduction ◽  
Order Reduction ◽  
Balance Equations ◽  
Model Order ◽  
Cad Systems ◽  
Computational Costs ◽  
Reduction Methods ◽  
Model Equations

The new algorithm of solving harmonic balance equations which used in electronic CAD systems is presented. The new algorithm is based on implementation to harmonic balance equations the ideas of model order reduction methods. This algorithm allows significantly reduce the size of memory for storing of model equations and reduce of computational costs.

Experimental and theoretical investigation of an equation of dendritic crystal growth

1991 ◽  
Vol 2 (3) ◽  
pp. 199-222
Author(s):  
J. N. Dewynne ◽  
F. N. H. Robinson
Keyword(s):  
Harmonic Balance ◽  
Large Time ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Dendritic Crystal ◽  
Method Of Harmonic Balance ◽  
Electronic Model ◽  
Wide Range ◽  
Periodic Form ◽  
Dissipative Terms

An experimental study using an analogue electronic model of the equation x‴ + x′ = є sin x, modified by the addition of small terms ax″ and βx with 0 < β < α ≫ є shows that these dissipative terms have a profound effect on the solutions for large time. If ∈ is not too large, experimental solutions tend to a simple periodic form, unlike the case α=β = 0. The existence of this limiting periodic form suggests the possibility of a simple analytic treatment using the method of harmonic balance, and this treatment leads to excellent agreement with the experimental results for a wide range of initial conditions and values of the parameters. The approach towards attracting limiting periodic solutions is analysed by using the method of multiple scales.

Symbolic computation of harmonic balance equations

1997 ◽  
Vol 68 (3) ◽  
pp. 449-460 ◽  
Author(s):  
J. C. Peyton ◽  
Jones M. Zhuang ◽  
I. Cankaya
Keyword(s):  
Symbolic Computation ◽  
Harmonic Balance ◽  
Balance Equations

scholarly journals APPROXIMATE SOLUTIONS OF DAMPED NON LINEAR SYSTEM WITH VARYING PARAMETER AND DAMPING FORCE

2019 ◽  
pp. 13-18
Author(s):  
REZAUL KARIM ◽  
PINAKEE DEY ◽  
SOMI AKTER ◽  
MOHAMMAD ASIF AREFIN ◽  
SAIKH SHAHJAHAN MIAH
Keyword(s):  
Approximate Solution ◽  
Dynamical Systems ◽  
Linear System ◽  
Harmonic Balance ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Damping Force ◽  
Kbm Method ◽  
Non Linear System

The study of second-order damped nonlinear differential equations is important in the development of the theory of dynamical systems and the behavior of the solutions of the over-damped process depends on the behavior of damping forces. We aim to develop and represent a new approximate solution of a nonlinear differential system with damping force and an approximate solution of the damped nonlinear vibrating system with a varying parameter which is based on Krylov–Bogoliubov and Mitropolskii (KBM) Method and Harmonic Balance (HB) Method. By applying these methods we solve and also analyze the finding result of an example. Moreover, the solutions are obtained for different initial conditions, and figures are plotted accordingly where MATHEMATICA and C++ are used as a programming language.

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