On the Dynamics of Elastohydrodynamic Mixed Lubricated Ball Bearings. Part II: Non-Linear Structural Vibration

2005 ◽  
Vol 219 (6) ◽  
pp. 423-433 ◽  
Author(s):  
M Sarangi ◽  
B. C. Majumdar ◽  
A. S. Sekhar
Keyword(s):  
Contact Stiffness ◽  
Equations Of Motion ◽  
Structural Vibration ◽  
Outer Race ◽  
Damping Coefficients ◽  
Lagrange’S Equation ◽  
Non Linear ◽  
Lubricated Contact ◽  
The Individual ◽  
Stiffness And Damping

Equations of motion of a ball bearing are formulated in generalized coordinates, using Lagrange's equation considering the vibrational characteristics of the individual constituents such as inner race, outer race, cage, and balls, in order to investigate the structural vibration of the bearing. This article is the second part of the present study dealing with structural vibration, whereas in the first part, elastohydrodynamic mixed lubricated contact stiffness and damping coefficients are determined. Utilizing these stiffness and damping coefficients, a non-linear load-deflection contact model is developed. This is then used in the equations of motion. The equations of motion are solved using Runge-Kutta integration technique. This work differs from the previous studies in the sense that the model simulates the vibration, considering that both the lubricated contact stiffness and damping correspond to the conservative and dissipative energies, respectively. It is observed that under undamped conditions, all the elements of the bearing actively participate in energy sharing and oscillate periodically, containing more than one frequency. The system vibration, however, died down rapidly in the presence of damping.

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.

Numerical study on the dynamic characteristics of water-lubricated rubber bearing under asperity contact

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Guo Xiang ◽  
Yijia Wang ◽  
Cheng Wang ◽  
Zhongliang Lv
Keyword(s):  
Elastic Modulus ◽  
Contact Stiffness ◽  
Water Film ◽  
Radial Clearance ◽  
Asperity Contact ◽  
Stiffness Coefficient ◽  
Content Type ◽  
Damping Coefficients ◽  
Rubber Bearing ◽  
Stiffness And Damping

Purpose In this study, the dynamic characteristics of the water-lubricated rubber bearing considering asperity contact are numerically studied, including water-film stiffness and damping coefficients and plastic-elastic contact stiffness coefficient. Design/methodology/approach The Kogut-Etsion elastic-plastic contact model is applied to calculate the contact stiffness coefficient at the bearing-bush interface and the perturbed method is used to calculate the stiffness and damping coefficients of water-film. In addition, the rubber deformation is determined by the finite element method (FEM) during the simulation. Parametric studies are conducted to assess the effects of the radial clearance, rubber thickness and elastic modulus on the dynamic characteristic of water-lubricated rubber bearing. Findings Numerical results indicate that stiffness and damping coefficients of water film and the contact stiffness of asperity are increased with the decreasing of the radial clearance and the dynamic coefficients are less sensitive to the rubber thickness compared with the elastic modulus of rubber. Furthermore, due to the existed groove, a sudden change of the water film direct stiffness and damping coefficients is observed when the eccentricity ratio ranges from 0.6 to 1.0. Originality/value It is expected this study can provide more information to establish a dynamic equation of water-lubricated rubber bearings exposed to mixed lubrication conditions.

Plain and Full Floating Bearing Simulations With Rigid Shaft Dynamics

2007 ◽  
Author(s):  
Ian McLuckie ◽  
Scott Barrett
Keyword(s):  
System Development ◽  
Equations Of Motion ◽  
Time Integration ◽  
Forced Response ◽  
Design Parameters ◽  
Integration Scheme ◽  
Damping Coefficients ◽  
Time Integration Scheme ◽  
Combined Solution ◽  
Stiffness And Damping

This paper shows a promising predictive bearing model that can be used to reduce turbocharger bearing system development times. Turbocharger development is normally done by varying design parameters such as bearing geometry in a very time consuming experimentation process. Full Floating Bearings (FFB) are used in most automotive turbochargers and, due to emissions regulations, there has been a push towards downsizing engines and applying turbo charging to generate optimized engine solutions for both gasoline and diesel applications. In this paper the turbocharger rotor is regarded as being rigid, and the equations of motion are solved using the Bulirsch Stoer time integration scheme. These equations are solved simultaneously with the bearing model which is used also to determine nonlinear stiffness and damping coefficients. The bearings are solved using a Rigid Hydro Dynamic (RHD) Finite Difference Successive Over Relaxation (SOR) scheme of Reynolds equation that includes both rotational and squeeze velocity terms. However the solver can also consider bearing and rotor elasticity in a Multi-Body Dynamic (MBD) and Elasto-Hydro Dynamic (EHD) combined solution. Two bearing types have been studied, a plain grooved (PGB) and a full floating bearing (FFB) for comparative purposes. The mathematical models used are generic and suitable for whole engine bearing studies. The results in this paper show they are suitable for determining the onset of turbocharger bearing instability, and also the means by which bearing instability may be suppressed. The current study has investigated forced response with the combined effects of gravity and unbalance. It is worth noting that the effects of both housing excitation and aerodynamic excitation from the compressor and turbine can be easily accommodated, and will be the subject of a future paper. Other topics introduced here that will be explored further in the future include the effect of bearing and rotor flexibility in the MBD and EHD solution and the use of automatically generated stiffness and damping coefficients for any bearing geometry.

Stability and Dynamics of Non-Linear Slider Behaviour Due to Disk Waviness in the Presence of Intermolecular and Electrostatic Forces

2005 ◽  
Author(s):  
Vineet Gupta ◽  
David B. Bogy
Keyword(s):  
Resonance Frequency ◽  
Equations Of Motion ◽  
Disk Surface ◽  
Air Bearing ◽  
Electrostatic Forces ◽  
Disk Interface ◽  
Non Linear ◽  
Bearing Stiffness ◽  
Stiffness And Damping ◽  
Nonlinear Nature

In this paper we present a theoretical investigation of the stability and the dynamics of the non-linear behavior of a slider at very low head media spacing. A single DOF head disk interface (HDI) model, with constant air bearing stiffness and damping has been used to study the effect of disk waviness on the nonlinear slider dynamics in the presence of intermolecular and electrostatic forces. A variational approach based on the principle of least action was used to derive the equations of motion of the slider. Further, a stability criteria was derived that helped to better understand the instabilities that appear in slider when the slider is flying in close proximity to the disk surface. Due to extremely nonlinear nature of the interaction between the slider and the disk, we observed some strange features of the motion of the slider. In particular the effects of the nonlinear interaction force, air bearing stiffness and damping on the instabilities of the periodic motions of the slider are discussed in detail. We found that the branch associated to the disk waviness frequencies larger than the resonance frequency is always stable and the branch associated to the disk waviness frequencies smaller than the resonance frequency exhibits two stable domains and one unstable domain. This analysis was further extended to include the nonlinear nature of air bearing stiffness and damping as well as contact at the HDI.

A Derivation of Stiffness and Damping Coefficients for Short Hydrodynamic Journal Bearings with Pseudo-Plastic Lubricants

2020 ◽  
Vol 36 (6) ◽  
pp. 943-953
Author(s):  
Zhuxin Tian ◽  
Runchang Chen
Keyword(s):  
Perturbation Method ◽  
Small Perturbation ◽  
Integral Formula ◽  
Journal Bearings ◽  
Damping Coefficients ◽  
Non Linear ◽  
Small Perturbation Method ◽  
Plastic Lubricants ◽  
Factor Α ◽  
Stiffness And Damping

ABSTRACTA new derivation considering the non-linear terms has been proposed to calculate stiffness and damping coefficients for short hydrodynamic journal bearings lubricated with pseudo-plastic fluids. The proposed method has relaxed the constraint of small perturbation method applicable to only small values of non-Newtonian factor α. An analytical solution is also given. The non-linear Reynolds equation is solved with a more reasonable boundary condition ∂p*/∂z* = 0 at the location of z*=0 while the analytical pressure distribution is obtained by seven-point Gauss-Legendre integral formula. When the non-dimensional non-Newtonian factor α is small, the stiffness and damping coefficients of computed by the proposed method can agree well with those from small perturbation method, which could verify the proposed derivation. As for large non-dimensional non-Newtonian factor α, the stiffness coefficients $K_{XX}^*$ , $K_{XY}^*$ and $K_{YX}^*$ as well as the damping coefficients $C_{XX}^*$ , $C_{XY}^*$ and $C_{YX}^*$ decrease with the increasing of non-dimensional non-Newtonian factor α. The significance of the derivation is that it can relax the constraint of small α and simplify the computation process.

scholarly journals Bifurcation analysis of rotor/bearing system using third-order journal bearing stiffness and damping coefficients

2021 ◽  
Author(s):  
T. A. El-Sayed ◽  
Hussein Sayed
Keyword(s):  
Equations Of Motion ◽  
Journal Bearings ◽  
Third Order ◽  
Damping Coefficients ◽  
The Third ◽  
Rotor Bearing ◽  
Bearing Stiffness ◽  
Rotor Stiffness ◽  
Stiffness And Damping ◽  
Bearing System

AbstractHydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

An Energy Approach to Linearizing Squeeze-Film Damper Forces

1985 ◽  
Vol 199 (1) ◽  
pp. 57-63
Author(s):  
E J Hahn
Keyword(s):  
Parametric Design ◽  
Squeeze Film ◽  
Transient Solution ◽  
Equivalent Stiffness ◽  
Design Studies ◽  
Damping Coefficients ◽  
Squeeze Film Dampers ◽  
Non Linear ◽  
Linear Elements ◽  
Stiffness And Damping

Analyses of multi-degree of freedom rotor-bearing systems incorporating non-linear elements, such as squeeze-film dampers, generally necessitate time consuming transient solution. Consequently, it is often too expensive to carry out parametric design studies on such systems. This paper presents a general technique for linearizing the non-linear element forces using equivalent stiffness and damping coefficients with energy dissipation and energy storage-release concepts. The approach is illustrated and tested for both centrally preloaded squeeze-film dampers and for squeeze-film dampers without centralizing springs under a combination of unidirectional and unbalance loading. The results predicted by using such equivalent stiffness and damping coefficients agree quite well with those obtained from the full transient solution, even where the unidirectional load exceeds the dynamic load and the damper is operating at high eccentricity. An iterative procedure is proposed which, with the aid of such stiffness and damping coefficients, should significantly reduce the computation time presently needed to carry out parametric design studies on general multi-degree of freedom systems incorporating non-linear elements such as squeeze-film dampers.

Vibration Control of a Rigid-Flexible Satellite During Attitude Maneuver

1999 ◽  
Author(s):  
Luiz Carlos Gadelha DeSouza ◽  
Silmara Alexandra DaSilva
Keyword(s):  
Vibration Control ◽  
Equations Of Motion ◽  
Body Motion ◽  
Structural Vibration ◽  
Control Law ◽  
Flexible Beam ◽  
Structural Vibration Control ◽  
Lagrange’S Equation ◽  
Attitude Maneuver ◽  
Flexible Part

Abstract The paper presents the results of an active structural vibration control of a flexible satellite performed by a proof-mass actuator (PMA) during an attitude rotation maneuver. The satellite investigated is composed of a rigid rub plus a cantilevered flexible beam with the PMA located at the beam free end. As the satellite maneuvers from rest to a pre-defined position, the rigid body motion can excite the flexible part of the satellite. The PMA tasks is to damp-out any residual vibration caused by this maneuver efficiently. The rigid/flexible satellite is modeled, using a relatively simple structural dynamics approach. Having found the vibration modes of the structure, expressions for kinetic and potential energy are derived. Lagrange’s equation is then applied to obtain the satellite equations of motion. The PMA gain selection is based oil an analytical approach which shows that the pole and zero of the fundamental mode is dominant. The efficiency of the PMA using velocity feedback with a PI control law was examined by numerical simulations for different control maneuvers strategies. It was shown that one such controller has damped the residual flexible vibration successfully. However, it was also shown that the control system efficiency is function of the maneuver strategy.

Optimization of the Individual Stiffness and Damping Parameters in Multiple-Tuned-Mass-Damper Systems

2005 ◽  
Vol 127 (1) ◽  
pp. 77-83 ◽  
Author(s):  
Lei Zuo ◽  
Samir A. Nayfeh
Keyword(s):  
Parameter Optimization ◽  
Substantial Improvement ◽  
Design Parameters ◽  
Feedback Gain ◽  
Primary System ◽  
Tuned Mass Dampers ◽  
H2 Control ◽  
Damping Coefficients ◽  
The Individual ◽  
Stiffness And Damping

The characteristics of multiple tuned-mass-dampers (MTMDs) attached to a single-degree-of-freedom primary system have been examined by many researchers. Several papers have included some parameter optimization, all based on restrictive assumptions. In this paper, we propose an efficient numerical algorithm to directly optimize the stiffness and damping of each of the tuned-mass dampers (TMDs) in such a system. We formulate the parameter optimization as a decentralized H2 control problem where the block-diagonal feedback gain matrix is composed of the stiffness and damping coefficients of the TMDs. The gradient of the root-mean-square response with respect to the design parameters is evaluated explicitly, and the optimization can be carried out efficiently. The effects of the mass distribution, number of dampers, total mass ratio, and uncertainties in system parameters are studied. Numerical results indicate that the optimal designs have neither uniformly spaced tuning frequencies nor identical damping coefficients, and that optimization of the individual parameters in the MTMD system yields a substantial improvement in performance. We also find that the distribution of mass among the TMDs has little impact on the performance of the system provided that the stiffness and damping can be individually optimized.

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