Dynamical Behavior of a Rotor Under Rotational Random Base Excitation

2007 ◽  
Author(s):  
Lucie Bachelet ◽  
Nicolas Driot ◽  
Guy Ferraris ◽  
Fabrice Poirion
Keyword(s):  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Gaussian White Noise ◽  
Forced Response ◽  
Power Spectrum Density ◽  
Largest Lyapunov Exponent ◽  
Base Excitation ◽  
Spectrum Density ◽  
The Stability ◽  
Truncated Gaussian

This paper investigates the dynamical behavior of a rotor under a random rotational base excitation, which is assumed to be a stationary and ergodic truncated Gaussian white noise. As the base motion is a rotation, the equations of motion present both internal and external random excitations. The stability of the rotor is then studied by computing the largest Lyapunov exponent with an iterative formula. Then, the power spectrum density of the stationary forced response is obtained from a Monte Carlo simulation. Finally, we perform a comparative analysis on the influence of the required number of modal shapes to describe accurately the response.

Stability and Stationary Response of a Skew Jeffcott Rotor With Geometric Uncertainty

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Nicolas Driot ◽  
Alain Berlioz ◽  
Claude-Henri Lamarque
Keyword(s):  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Forced Response ◽  
Stochastic Methods ◽  
Steady State Response ◽  
State Response ◽  
Internal Excitation ◽  
Geometric Uncertainty ◽  
Jeffcott Rotor ◽  
Support Motion

The aim of this work is to apply stochastic methods to investigate uncertain parameters of rotating machines with constant speed of rotation subjected to a support motion. As the geometry of the skew disk is not well defined, randomness is introduced and affects the amplitude of the internal excitation in the time-variant equations of motion. This causes uncertainty in dynamical behavior, leading us to investigate its robustness. Stability under uncertainty is first studied by introducing a transformation of coordinates (feasible in this case) to make the problem simpler. Then, at a point far from the unstable area, the random forced steady state response is computed from the original equations of motion. An analytical method provides the probability of instability, whereas Taguchi’s method is used to provide statistical moments of the forced response.

Stability and Stationary Response of a Skew Jeffcott Rotor With Geometric Uncertainty

2007 ◽  
Author(s):  
Nicolas Driot ◽  
Alain Berlioz ◽  
Claude Henri Lamarque
Keyword(s):  
Translational Motion ◽  
Dynamical Behavior ◽  
Random Parameter ◽  
Forced Response ◽  
Motion Equations ◽  
State Response ◽  
Geometric Uncertainty ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Original Motion

The dynamical behavior of an asymmetrical Jeffcott rotor subjected to a base translational motion is investigated. As the geometry of the skew disk is not well defined, we introduce some randomness. This uncertainty affects a particular parameter in the time-variant motion equations. Consequently, the amplitude of the parametric excitation is a random parameter which leads us to investigate the robustness of the dynamics. The stability is first studied by introducing a transformation of coordinates (feasible in this case) making the problem simpler. Then, far away from the unstable area, the random forced steady state response is computed from the original motion equations. The Taguchi’s method is used to provide statistical moments of the forced response.

Analytical Study on a Dynamical Behavior of Loading Buckets Interacting With Bulk Materials

2003 ◽  
Author(s):  
Mitsuhiro Yoshida ◽  
Shigeki Morii
Keyword(s):  
Analytical Model ◽  
Analytical Study ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Dynamical Behavior ◽  
Bulk Materials ◽  
Cross Term ◽  
Linear Equations Of Motion ◽  
Scale Experiment ◽  
The Stability

A new analytical model has been developed to calculate the dynamical behavior of a column on which buckets loading and interacting with bulk materials, such as iron ore and coal with various grain sizes, are set. A set of linear equations of motion was obtained by considering the effective load, the stiffness, and the friction of materials. From the analysis of the equation, the growth of vibration in a column was indicated, which was caused by the cross term of a stiffness matrix because of the interaction between buckets and a material. In addition, bucket speed was found to affect the damping of the vibration through the critical condition Δxω∼vB where Δx is bucket displacement, ω is the characteristic frequency of a column, and v is bucket speed. A simplified characteristic function was obtained for the stability of a system and compared with the results of a laboratory-scale experiment in order to evaluate the analytical model.

A Parametric Analysis of the Transient Forced Response of Noncontacting Coned-Face Gas Seals

2001 ◽  
Vol 124 (1) ◽  
pp. 151-157 ◽  
Author(s):  
Itzhak Green ◽  
Roger M. Barnsby
Keyword(s):  
Equations Of Motion ◽  
Forced Response ◽  
Coupled Analysis ◽  
Face Seal ◽  
Reference Case ◽  
Stable Mode ◽  
Pressure Drops ◽  
Stability Issue ◽  
Gas Seals ◽  
The Stability

A properly designed mechanical face seal must satisfy two requirements: (1) the seal must be stable, and (2) the seal forced response must be such that the stator tracks the misaligned rotor with the smallest clearance possible, with the smallest relative tilt, and with the largest minimum film thickness. The stability issue was investigated in a previous paper. Here a numerical solution is presented for the transient response of a noncontacting gas lubricated face seal that is subjected to stator and rotor forcing misalignments. The seal dynamic response is obtained in axial and angular modes of motion in a coupled analysis where the Reynolds equation and the equations of motion are solved simultaneously. The steady-state response is first identified for a reference case. Subsequently a parametric study is performed to gauge the influence of the various seal effects, such as speeds, inner to outer radii ratios, face coning heights, pressure drops, support stiffness and damping, and forcing misalignments. The transient responses to static stator misalignment and rotor runout are given, showing that properly designed coned face seals can operate in a stable mode with the stator tracking dynamically a misaligned rotor.

Orientation holes positioning of printed board based on LS-Power spectrum density algorithm

2018 ◽  
Vol 35 (3-4) ◽  
pp. 277-288
Author(s):  
Xiaxia ZENG ◽  
Zhenhua SONG ◽  
Wenzhong LIN ◽  
Haibo LUO
Keyword(s):  
Power Spectrum ◽  
Power Spectrum Density ◽  
Spectrum Density

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

2021 ◽  
pp. 107754632199358
Author(s):  
Ali Fasihi ◽  
Majid Shahgholi ◽  
Saeed Ghahremani
Keyword(s):  
Energy Harvester ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Nonlinear Energy Sink ◽  
Piezoelectric Energy ◽  
Energy Sink ◽  
Two Degree Of Freedom ◽  
Nonlinear Energy

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

Random Response Analysis of a Large-Size Cooling Tower Subjected to the Stationary Earthquake Motion

2013 ◽  
Vol 423-426 ◽  
pp. 1589-1593
Author(s):  
Jia Ning Zhu ◽  
Ya Zhou Xu ◽  
Guo Liang Bai ◽  
Rui Wen Li
Keyword(s):  
Power Spectrum ◽  
Random Vibration ◽  
Cooling Tower ◽  
Response Spectrum ◽  
Power Spectrum Density ◽  
Random Response ◽  
Large Size ◽  
Earthquake Motion ◽  
Spectrum Density ◽  
Random Vibration Theory

The response of a large-size cooling tower with 250m high subjected to the seismic action are investigated by both random vibration theory and response spectrum method. Shell element is taken to model the tower body, and beam element is used for the circular foundation and supporting columns. The earthquake motion input is a colored filtered white noise model and mode superposition method is adopted to analyze the random response of the large-size cooling tower. The paper presents the power spectrum density functions (PDF) and standard deviation of the displacement of the top and characteristic node, and the analysis results indicate that the results of the stationary random vibration theory and the response spectrum method are the same order of magnitude. The power spectrum density function of the bottom node stress is obviously bigger than the one at the top and the throat, and the random response of meridonal stress is dominated at the top. In addition, the peak frequency position of the power spectrum density function is different from the corresponding stress.

A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions

2016 ◽  
Vol 26 (13) ◽  
pp. 1650226 ◽  
Author(s):  
Eduardo M. A. M. Mendes ◽  
Erivelton G. Nepomuceno
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Equations Of Motion ◽  
Original System ◽  
Glass System ◽  
Largest Lyapunov Exponent ◽  
Simple Method ◽  
Interval Extensions

In this letter, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. The exponent is estimated from the slope of the line derived from the lower bound error when considering two interval extensions of the original system. It is shown that the algorithm is robust, fast and easy to implement and can be considered as alternative to other algorithms available in the literature. The method has been successfully tested in five well-known systems: Logistic, Hénon, Lorenz and Rössler equations and the Mackey–Glass system.

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