scholarly journals Application of harmonic balance method for solving non-linear equations of motion

2018 ◽  
Vol 182 ◽  
pp. 02024
Author(s):  
Robert Kostek
Keyword(s):  
Harmonic Balance ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Harmonic Balance Method ◽  
Time History ◽  
Balance Method ◽  
Non Linear ◽  
Time Histories ◽  
History Of ◽  
Non Linear Equations

This article presents the advantages and limitations of a harmonic balance method applied for solving non-linear equations of monition. This method provides an opportunity to find stable and unstable periodic solutions, which was demonstrated for a few equations. An error of solution decreases rapidly with increase of number of harmonics for smooth time history of acceleration, which shows convergence; whereas, for discontinuous time histories, this method is not effective.

scholarly journals Multi-parametric Dynamic Analysis of a Rolling Bearings System

2021 ◽  
Vol 67 (9) ◽  
pp. 421-432
Author(s):  
Almatbek Kydyrbekuly ◽  
Gulama-Garip Alisher Ibrayev ◽  
Tangat Ospan ◽  
Anatolij Nikonov
Keyword(s):  
Harmonic Balance ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Stiffness Coefficient ◽  
Rolling Bearings ◽  
Linear Characteristic ◽  
Non Linear ◽  
Linear Equations Of Motion ◽  
Oscillation Frequencies ◽  
Non Linear Equations

A method for calculating amplitudes and constructing frequency characteristics of forced and self-excited vibrations of a rotor-fluid-foundation system on rolling bearings with a non-linear characteristic based on the method of complex amplitudes and harmonic balance has been developed. Non-linear equations of motion of the rotor-fluid-foundation system are derived, and analytical methods of their solution are presented. Frequencies of fundamental and ultra-harmonic resonances are determined. The intervals between self-oscillation frequencies are estimated. The dependence of amplitudes on the amount of fluid in the rotor cavity, the mass of the foundation, linear imbalance, the value of the stiffness coefficient, and the damping coefficient is shown.

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.

Simulations of Unsteady Blade Row Interactions Using Linear and Non-Linear Frequency Domain Methods

2015 ◽  
Author(s):  
Christian Frey ◽  
Graham Ashcroft ◽  
Hans-Peter Kersken
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Measurement Data ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Frequency Domain Methods ◽  
Blade Row ◽  
Non Linear ◽  
Rotor Stator Interaction ◽  
Unsteady Simulation

This paper compares various approaches to simulate unsteady blade row interactions in turbomachinery. Unsteady simulations of turbomachinery flows have gained importance over the last years since increasing computing power allows the user to consider 3D unsteady flows for industrially relevant configurations. Furthermore, for turbomachinery flows, the last two decades have seen considerable efforts in developing adequate CFD methods which exploit the rotational symmetries of blade rows and are therefore up to several orders of magnitude more efficient than the standard unsteady approach for full wheel configurations. This paper focusses on the harmonic balance method which has been developed recently by the authors. The system of equations as well as the iterative solver are formulated in the frequency domain. The aim of this paper is to compare the harmonic balance method with the time-linearized as well as the non-linear unsteady approach. For the latter the unsteady flow fields in a fan stage are compared to reference results obtained with a highly resolved unsteady simulation. Moreover the amplitudes of the acoustic modes which are due to the rotor stator interaction are compared to measurement data available for this fan stage. The harmonic balance results for different sets of harmonics in the blade rows are used to explain the minor discrepancies between the time-linearized and unsteady results published by the authors in previous publications. The results show that the differences are primarily due to the neglection of the two-way coupling in the time-linearized simulations.

Dynamic Analysis of Structures With Nonlinear Bolted Joints by Using an Adaptive Harmonic Balance Method

2009 ◽  
Author(s):  
Vincent Jaumouille´ ◽  
Jean-Jacques Sinou
Keyword(s):  
Harmonic Balance ◽  
Strain Energy ◽  
Harmonic Balance Method ◽  
Bolted Joints ◽  
Balance Method ◽  
Continuation Methods ◽  
Computational Time ◽  
Bolted Joint ◽  
Non Linear ◽  
Linear Effects

Aeronautical structures are commonly assembled with bolted joints in which friction phenomena provide damping on the dynamic behaviour. Some models, mostly non linear, have consequently been developed and the harmonic balance method (HBM) is adapted to compute non linear response functions in the frequency domain. The basic idea is to develop the response as a Fourier series and to solve equations linking Fourier coefficients. One specific HBM feature is that response accuracy improves as the number of harmonics increases, at the expense of larger computational time. Thus the aim of this study is to develop an adaptive HBM which appreciates numerically the contribution of each harmonic on the dynamic response. For a given precision, the number of retained harmonics is adapted by an algorithm which integrates a numerical criterion based on an approximate strain energy. The application case is an asymmetrical two cantilever beam system linked by a bolted joint represented by a nonlinear LuGre model. Condensation and continuation methods are used to accelerate calculation. Adaptive HBM shows that, for a given value of the criterion, the number of harmonics may increase on resonances indicating that non linear effects are predominant.

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