Dynamic Response of an Unbalanced Rotor Supported on Bearing With Outer Race Waviness

2013 ◽  
Author(s):  
P. K. Kankar ◽  
Satish C. Sharma ◽  
S. P. Harsha
Keyword(s):  
Dynamic Response ◽  
Chaotic Behavior ◽  
Mathematical Formulation ◽  
Rotational Frequency ◽  
Hertzian Contact ◽  
Radial Clearance ◽  
Outer Race ◽  
Unbalanced Rotor ◽  
Non Linear ◽  
Rolling Elements

The paper investigates the non-linear dynamic response of an unbalanced rotor supported on ball bearings with outer race waviness. The excitation is due to unbalanced force and waviness on outer race. The sources of non-linearities are both the radial clearance as well as the Hertzian contact between races and rolling elements. The nonlinear responses due to unbalanced rotor supported on bearings are investigated. The combined effects like non-linear stiffness and non-linear damping for unbalanced rotor with bearing waviness have been considered and analyzed in detail for a rotor bearing system. In the mathematical formulation, the contacts between the rolling elements and the races are considered as an oscillating spring-mass-damper system. The appearance of regions of periodic, sub-harmonic and chaotic behavior is seen to be strongly dependent on the number of waves in the outer race. The results show the appearance of instability and chaos in the dynamic response as the number of waves in the outer race is changed. The study indicates that the interaction of ball passage frequency (ωbp) due to outer race waviness and rotational frequency (X) due to the unbalanced rotor force. Poincaré maps and frequency responses are used to elucidate and to illustrate the diversity of the system behavior.

scholarly journals Modeling and Dynamic Analysis of Spherical Roller Bearing with Localized Defects: Analytical Formulation to Calculate Defect Depth and Stiffness

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Behnam Ghalamchi ◽  
Jussi Sopanen ◽  
Aki Mikkola
Keyword(s):  
Dynamic Model ◽  
Contact Stiffness ◽  
Roller Bearing ◽  
Measurement Data ◽  
Contact Forces ◽  
Hertzian Contact ◽  
Radial Clearance ◽  
Outer Race ◽  
Rolling Elements ◽  
Localized Defects

Since spherical roller bearings can carry high load in both axial and radial direction, they are increasingly used in industrial machineries and it is becoming important to understand the dynamic behavior of SRBs, especially when they are affected by internal imperfections. This paper introduces a dynamic model for an SRB that includes an inner and outer race surface defect. The proposed model shows the behavior of the bearing as a function of defect location and size. The new dynamic model describes the contact forces between bearing rolling elements and race surfaces as nonlinear Hertzian contact deformations, taking radial clearance into account. Two defect cases were simulated: an elliptical surface on the inner and outer races. In elliptical surface concavity, it is assumed that roller-to-race-surface contact is continuous as each roller passes over the defect. Contact stiffness in the defect area varies as a function of the defect contact geometry. Compared to measurement data, the results obtained using the simulation are highly accurate.

scholarly journals A theoretical model for predicting the vibration response of outer race defective ball bearing

2018 ◽  
Vol 7 (2) ◽  
pp. 289
Author(s):  
Samir Shaikh ◽  
Sham Kulkarni
Keyword(s):  
Theoretical Model ◽  
Ball Bearing ◽  
Mathematical Formulation ◽  
Vibration Response ◽  
Contact Forces ◽  
Hertzian Contact ◽  
Outer Race ◽  
Rolling Element ◽  
Rolling Elements ◽  
Extended Type

The theoretical model with 2 degree-of-freedom system is developed for predicting the vibration response and analyze frequency properties in an extended type defective ball bearing. In the mathematical formulation, the contact between the races and rolling element considered as non-linear springs. The contact forces produced during the collaboration of rolling elements are obtained by utilizing Hertzian contact deformation hypothesis. The second order nonlinear differential equation of motion is solved using a state space variable method with the help of MATLAB software and the vibration acceleration response of the defective ball bearing presented in the frequency spectrum. The effects of variation in speed and size of the defect on characteristic frequency of extended fault on the outer raceway of the ball bearing have been investigated. The theoretical results of the healthy (non defective) and defective bearing are compared with each other.

Dynamic Analysis of Ball Bearings Due to Clearance Effect

2009 ◽  
Author(s):  
S. H. Upadhyay ◽  
S. C. Jain ◽  
S. P. Harsha
Keyword(s):  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Peak Frequency ◽  
System Response ◽  
Ball Bearings ◽  
Outer Race ◽  
Lagrange’S Equation ◽  
Rolling Elements ◽  
Machine Systems ◽  
The Individual

In this paper, the nonlinear dynamic behavior of ball bearings due to radial internal clearance and rotor speed has been analyzed. The approach presented in this paper accounts for the contact between rolling elements and inner/outer races. The equations of motion of a ball bearing are formulated in generalized coordinates, using Lagrange’s equation considering the vibration characteristics of the individual constitute such as inner race, outer race, rolling elements. The effects of speed of rotor in which rolling element bearings shows periodic, quasi-periodic and chaotic behavior are analyzed. The results also show the intermittent chaotic behavior in the dynamic response is seen to be strongly dependent on the speed of the rotor. The results are obtained in the form of frequency responses. The validity of the proposed model verified by comparison of frequency components of the system response with those obtained from experiments. The peak-to-peak frequency response of the system for each speed is obtained. The current study provides a powerful tool design and health monitoring of machine systems.

Intermittent Chaotic Dynamics of Rail Axle Supported by Roller Bearings

2009 ◽  
Author(s):  
S. P. Harsha ◽  
C. ‘Nat’ Nataraj
Keyword(s):  
High Speed ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Radial Clearance ◽  
High Speed Rail ◽  
Frequency Spectra ◽  
Roller Bearings ◽  
Nonlinear Springs ◽  
Rolling Elements

In this paper, intermittent chaotic analysis of high speed rail axle supported by roller bearings has been analyzed. In the analytical formulation, the contacts between rolling elements and races are considered as nonlinear springs, whose stiffness values are obtained by using Hertzian elastic contact deformation theory. The results show the appearance of instability and chaos in the dynamic response as the speed of the axle-bearing system is changed. Period doubling and mechanism of intermittency have been observed which lead to chaos. The appearance of regions of periodic, sub-harmonic and chaotic behavior is seen to be strongly dependent on the radial clearance. Poincare´ maps, time response and frequency spectra are used to elucidate and to illustrate the diversity of the system behavior.

The Effect of Flexible Pipe Non-Linear Bending Stiffness Behavior on Bend Stiffener Analysis

2007 ◽  
Author(s):  
Marcelo Caire ◽  
Murilo Augusto Vaz
Keyword(s):  
Mathematical Formulation ◽  
Bending Stiffness ◽  
Radial Clearance ◽  
Flexible Pipe ◽  
Element Analysis ◽  
Linear Ordinary Differential Equations ◽  
Analysis And Design ◽  
Extreme Load ◽  
Non Linear ◽  
Pipe Bending

Bend stiffeners are critical components for flexible risers and umbilical cables employed to ensure a safe transition at the riser-vessel interface, avoiding overbending and accumulation of high cyclic fatigue damage. The analysis and design of bend stiffeners usually consider the system as a unique beam, in which the pipe bending response is linear. However, the structural mechanics of these complex layered structures is governed by internal friction mechanisms that yield non-linear moment versus curvature relationship. In fact, the pipe structure exhibits an approximately bi-linear hysteretic bending moment against curvature relationship arising from the progressive activation of friction and consequential slipping between adjacent layers. The flexible pipe bending stiffness substantially reduces after a given critical curvature (i.e., after slip between adjacent layers) is reached. In this paper, the effect of this flexible pipe non-linear response on the bend stiffener design is evaluated. The mathematical formulation and the solution methodology are presented. A set of four non-linear ordinary differential equations is obtained from geometrical compatibility, equilibrium of forces and moments and constitutive equations and a numerical solution is obtained using the shooting method. A finite element analysis is developed to validate the analytical model and to assess the effect of the radial clearance between the structures on the bend stiffener response. A case study is presented for some static loading conditions and it is observed that the bending stiffness bi-linear behavior may not affect the bend stiffener extreme load design results, but it may significantly influence the fatigue analysis.

scholarly journals Vibration response of defect-ball-defect of rolling bearing with compound defects on both inner and outer races

2021 ◽  
Vol 1207 (1) ◽  
pp. 012006
Author(s):  
Wei Luo ◽  
Changfeng Yan ◽  
Junbao Yang ◽  
Yaofeng Liu ◽  
Lixiao Wu
Keyword(s):  
Rolling Bearing ◽  
Vibration Response ◽  
Radial Clearance ◽  
Rolling Bearings ◽  
Outer Race ◽  
Response Characteristics ◽  
Bearing Fault Diagnosis ◽  
Theoretical Criterion ◽  
Vibration Model ◽  
Rolling Elements

Abstract Aiming at the problem that the existing compound defects model of rolling bearings under radial load is difficult to reflect the actual contact between rolling elements and defects. A new model is proposed to accurately reflect the simultaneous or sequential contact between inner and outer race defects and rolling elements. Considering the coupled excitation between shaft and bearing and pedestal, time-varying displacement excitation, and radial clearance, a four degree-of-freedom vibration model of rolling bearing with compound faults on both inner and outer races is built. The vibration equations are calculated by the method of numerical way, and the model is verified by experiment. The vibration response characteristics of the Defect-Ball-Defect model are studied, which renders a theoretical criterion for bearing fault diagnosis.

Non-linear dynamic response analysis of cylindrical roller bearings due to rotational speed

2018 ◽  
Vol 233 (2) ◽  
pp. 379-390 ◽  
Author(s):  
Pravajyoti Patra ◽  
V Huzur Saran ◽  
SP Harsha
Keyword(s):  
Dynamic Behaviour ◽  
Hertzian Contact ◽  
Phase Portraits ◽  
Response Analysis ◽  
Roller Bearings ◽  
Contact Points ◽  
Unbalanced Rotor ◽  
Non Linear ◽  
The Impact ◽  
Cylindrical Roller

The dynamic behaviour of cylindrical roller bearings is presented, in both balanced and unbalanced conditions as a function of speed. The stiffness and damping non-linearities at the contact points (due to Hertzian contact force between rollers and races), radial internal clearance and unbalanced rotor force make the bearing system non-linear. Presently, the differential equations representing the dynamics of the cylindrical roller bearings have been obtained using Lagrange’s equation and solved numerically using modified Newmark-β method. The results of the analyses of various motion behaviours are presented as time–displacement responses, orbit plots, phase portraits, Poincaré maps and Fast Fourier Transform plots. The obtained responses revealed the sensitive behaviour of the system from periodic to quasi-periodic and chaotic with speed variations for both balanced and unbalanced rotor conditions. Also, intermittent chaotic behaviour has been observed. A pattern of the interaction between rotational and variable compliance vibration is observed with speed variations. The frequency pattern analysis (with different techniques used like phase/orbit plots and Poincaré plots) for healthy cylindrical bearing and different rotor conditions under different applied non-linearity consideration is a new attempt to analyse dynamic behaviour of the bearing. This analysis is helpful for online monitoring of fault-free cylindrical roller bearings and for studying the impact of speed on system’s dynamical behaviour.

Vibration Analysis of an Unbalanced Rotating Shaft Due to Ball Waviness

2009 ◽  
Author(s):  
S. H. Upadhyay ◽  
S. C. Jain ◽  
S. P. Harsha
Keyword(s):  
High Speed ◽  
Contact Stiffness ◽  
Nonlinear Differential Equations ◽  
Nonlinear Dynamic Analysis ◽  
Rotational Frequency ◽  
Hertzian Contact ◽  
Damping Force ◽  
Rotating Shaft ◽  
Unbalanced Rotor ◽  
Wave Passage

This paper presents an analytical model to investigate the nonlinear dynamic analysis of high speed ball bearings due to ball waviness and unbalanced rotor. The Hertzian contact theory is applied to calculate the elastic deflection and nonlinear contact force, while the rotor has translational and angular motions. In the analytical formulation, the contacts between rolling elements and inner/outer races are considered as nonlinear springs. A nonlinear damping formula, correlating the contact damping force with the equivalent contact stiffness and contact deformation rate is used in the derivation. The numerical integration technique Newmark-β with Newton-Raphson method is used to solve the nonlinear differential equations iteratively. The results are presented in the form of FFTs. The formulation predicts the discrete spectra with specific frequency components for each order of ball waviness. Numerical results obtained from the simulation are validated with respect to those of prior results. Due to an unbalanced rotor, the system is bi-periodically excited. In the vibration spectrum, the peak amplitude of vibrations appears at the wave passage frequency (Nwωroll), rotational frequency (X) and interaction between wave passage and rotational frequencies.

Non-linear seismic dynamic response of curved steel bridges equipped with LRB supports

2010 ◽  
Vol 3 (1) ◽  
pp. 34-41 ◽  
Author(s):  
Carlos Mendez Galindo ◽  
Javier Gil Belda ◽  
Toshiro Hayashikawa
Keyword(s):  
Dynamic Response ◽  
Steel Bridges ◽  
Non Linear ◽  
Curved Steel
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