scholarly journals Multi-Dimensional Harmonic Balance With Uncertainties Applied to Rotor Dynamics

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
J. Didier ◽  
J.-J. Sinou ◽  
B. Faverjon
Keyword(s):  
Monte Carlo ◽  
Dynamic Systems ◽  
Harmonic Balance ◽  
Polynomial Chaos ◽  
Harmonic Balance Method ◽  
Numerical Approach ◽  
Rotor Dynamics ◽  
Geometrical Parameters ◽  
Chaos Expansion ◽  
Response Functions

This paper describes the coupling of a Multi-Dimensional Harmonic Balance Method (MHBM) with a Polynomial Chaos Expansion (PCE) to determine the dynamic response of quasi-periodic dynamic systems subjected to multiple excitations and uncertainties. The proposed method will be applied to a rotor system excited at its support. Uncertainties considered include both material and geometrical parameters as well as excitation sources. To demonstrate the effectiveness and validity of the proposed numerical approach, the results that include mean, variation of the response, envelopes of the Frequency Response Functions and orbits will be systematically compared to a classical Monte Carlo approach.

scholarly journals Stochastic Modeling of Electrical Field in Potato Tuber using Polynomial Chaos Expansion

2019 ◽  
Vol 29 ◽  
pp. 01008
Author(s):  
Bartosz Sawicki ◽  
Artur Krupa
Keyword(s):  
Monte Carlo ◽  
Potato Tuber ◽  
Polynomial Chaos ◽  
Ohmic Heating ◽  
Stochastic Approach ◽  
Polynomial Chaos Expansion ◽  
Chaos Expansion ◽  
Computational Power ◽  
Electrical Field ◽  
Stochastic Parameters

The paper deals with numerical modeling of objects with a natural origin. The stochastic approach based on description using random variables allows processing such challenges. The Monte-Carlo methods are known a tool for simulations containing stochastic parameters however, they require significant computational power to obtain stable results. Authors compare Monte- Carlo with more advanced Polynomial Chaos Expansion (PCE) method. Both statistical tools have been applied for simulation of the electric field used in ohmic heating of potato tuber probes. Results indicate that PCE is remarkably faster, however, it simplifies some probabilistic features of the solution.

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.

On the convergence of the quasi-regression method: polynomial chaos and regularity

2017 ◽  
Vol 54 (2) ◽  
pp. 424-443
Author(s):  
Je Guk Kim
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Rate Of Convergence ◽  
Polynomial Chaos ◽  
Polynomial Chaos Expansion ◽  
Regression Method ◽  
Basis Functions ◽  
Chaos Expansion ◽  
Value Functions ◽  
Weaker Conditions

Abstract We present an analysis of convergence of a quasi-regression Monte Carlo method proposed by Glasserman and Yu (2004). We show that the method surely converges to the true price of an American option even under multiple underlyings via polynomial chaos expansion and weaker conditions than those used in Glasserman and Yu (2004). Further, we show the number of simulation paths grows exponentially in the number of basis functions to obtain convergence in implementing the method. Finally, we propose a rate of convergence considering regularity of value functions.

A Study of the Influence of Static Eccentricity and Unbalance on Cavitation Effects in a Squeeze Film Damped Flexible Rotor

2002 ◽  
Author(s):  
Philip Bonello ◽  
Michael J. Brennan ◽  
Roy Holmes
Keyword(s):  
Dynamic Systems ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Squeeze Film ◽  
Flexible Rotor ◽  
Test Rig ◽  
Squeeze Film Damper ◽  
Static Eccentricity ◽  
Rotor Dynamic ◽  
Rigid Rotors

The study of eccentric squeeze film damped rotor dynamic systems has largely concentrated on rigid rotors. In this paper, a newly developed receptance harmonic balance method is used to efficiently analyze a squeeze film damped flexible rotor test rig. The aim of the study is to investigate the influence of damper static eccentricity and unbalance level on cavitation and its resulting effect on the vibration level. By comparing predictions for the rotor vibration levels obtained respectively with, and without, lower pressure limits for the eccentric squeeze film damper model, it is demonstrated that cavitation is promoted by increasing static eccentricity and/or unbalance level. This, in turn, is found to have a profound effect on the predictions for the critical vibration levels, which such dampers are designed to attenuate. The reported findings are backed by experimental evidence from the test rig.

Periodicity and Stability in Transverse Motion of a Nonlinear Rotor-Bearing System Using Generalized Harmonic Balance Method

2016 ◽  
Author(s):  
Zhiwei Liu ◽  
Yuefang Wang
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Numerical Approach ◽  
Iteration Scheme ◽  
Balance Method ◽  
Transverse Motion ◽  
Harmonic Load ◽  
Rotor Bearing ◽  
Generalized Harmonic Balance Method ◽  
Bearing System

Many rotor assemblies of industrial turbo-machines are supported by oil-lubricated bearings. It is well known that the operation safety of these machines is highly dependent on rotors whose stability is closely related to the whirling motion of lubricant oil. In this paper, the problem of transverse motion of rotor systems considering bearing nonlinearity is revisited. A symmetric, rigid Jeffcott rotor is modeled considering unbalanced mass and short bearing forces. A semi-analytical, semi-numerical approach is presented based on the Generalized Harmonic Balance method and the Newton-Raphson iteration scheme. The external load of the system is decomposed into a Fourier series with multiple harmonic loads. The amplitude and phase with respect to each harmonic load are solved iteratively. The stability of the motion response is analyzed through identification of eigenvalues at the fixed point mapped from the linearized system using harmonic amplitudes. The solutions of the present approach are compared to the ones from time-domain numerical integrations using the Runge-Kutta method and they are found in good agreement for stable periodic motions. It is revealed through bifurcation analysis that evolution of the motion in the nonlinear rotor-bearing system is complicated. The Hopf bifurcation of synchronous vibration represents the start of the oil whirl. The phenomenon of oil whip is identified when the saddle-node bifurcation of sub-synchronous vibration takes place.

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