Dynamic Modeling and Vibration Response Simulation for High Speed Rolling Ball Bearings With Localized Surface Defects in Raceways

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Linkai Niu ◽  
Hongrui Cao ◽  
Zhengjia He ◽  
Yamin Li
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Surface Defects ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Rolling Ball ◽  
Vibration Responses

A dynamic model is developed to investigate vibrations of high speed rolling ball bearings with localized surface defects on raceways. In this model, each bearing component (i.e., inner raceway, outer raceway and rolling ball) has six degrees of freedom (DOFs) to completely describe its dynamic characteristics in three-dimensional space. Gyroscopic moment, centrifugal force, lubrication traction/slip between bearing component are included owing to high speed effects. Moreover, local defects are modeled accurately and completely with consideration of additional deflection due to material absence, changes of Hertzian contact coefficient and changes of contact force directions due to raceway curvature variations. The obtained equations of motion are solved numerically using the fourth order Runge–Kutta–Fehlberg scheme with step-changing criterion. Vibration responses of a defective bearing with localized surface defects are simulated and analyzed in both time domain and frequency domain, and the effectiveness of fault feature extraction techniques is also discussed. An experiment is carried out on an aerospace bearing test rig. By comparing the simulation results with experiments, it is confirmed that the proposed model is capable of predicting vibration responses of defective high speed rolling ball bearings effectively.

Detection of Cracks in Mistuned Bladed Disks Using Reduced-Order Models and Vibration Data

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Chulwoo Jung ◽  
Akira Saito ◽  
Bogdan I. Epureanu
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Three Dimensional ◽  
Cost Savings ◽  
Forced Response ◽  
Response Analysis ◽  
Reduced Order Models ◽  
Reduced Order

A novel methodology to detect the presence of a crack and to predict the nonlinear forced response of mistuned turbine engine rotors with a cracked blade and mistuning is developed. The combined effects of the crack and mistuning are modeled. First, a hybrid-interface method based on component mode synthesis is employed to develop reduced-order models (ROMs) of the tuned system with a cracked blade. Constraint modes are added to model the displacements due to the intermittent contact between the crack surfaces. The degrees of freedom (DOFs) on the crack surfaces are retained as active DOFs so that the physical forces due to the contact/interaction (in the three-dimensional space) can be accurately modeled. Next, the presence of mistuning in the tuned system with a cracked blade is modeled. Component mode mistuning is used to account for mistuning present in the uncracked blades while the cracked blade is considered as a reference (with no mistuning). Next, the resulting (reduced-order) nonlinear equations of motion are solved by applying an alternating frequency/time-domain method. Using these efficient ROMs in a forced response analysis, it is found that the new modeling approach provides significant computational cost savings, while ensuring good accuracy relative to full-order finite element analyses. Furthermore, the effects of the cracked blade on the mistuned system are investigated and used to detect statistically the presence of a crack and to identify which blade of a full bladed disk is cracked. In particular, it is shown that cracks can be distinguished from mistuning.

A New Approach to Modeling Surface Defects in Bearing Dynamics Simulations

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Ankur Ashtekar ◽  
Farshid Sadeghi ◽  
Lars-Erik Stacke
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Surface Defects ◽  
Equations Of Motion ◽  
Infinite Domain ◽  
Step Size ◽  
Deep Groove ◽  
Dynamics Simulations ◽  
Bearing Dynamics

A dynamic model for deep groove and angular contact ball bearings was developed to investigate the influence of race defects on the motions of bearing components (i.e., inner and outer races, cage, and balls). In order to determine the effects of dents on the bearing dynamics, a model was developed to determine the force-deflection relationship between an ellipsoid and a dented semi-infinite domain. The force-deflection relationship for dented surfaces was then incorporated in the bearing dynamic model by replacing the well-known Hertzian force-deflection relationship whenever a ball/dent interaction occurs. In this investigation, all bearing components have six degrees-of-freedom. Newton’s laws are used to determine the motions of all bearing elements, and an explicit fourth-order Runge–Kutta algorithm with a variable or constant step size was used to integrate the equations of motion. A model was used to study the effect of dent size, dent location, and inner race speed on bearing components. The results indicate that surface defects and irregularities like dent have a severe effect on bearing motion and forces. Furthermore, these effects are even more severe for high-speed applications. The results also demonstrate that a single dent can affect the forces and motion throughout the entire bearing and on all bearing components. However, the location of the dent dictates the magnitude of its influence on each bearing component.

Experimental and numerical analyses of wedge effects on the rooster tail and porpoising phenomenon of a high-speed planing craft in calm water

2019 ◽  
Vol 233 (13) ◽  
pp. 4637-4652 ◽  
Author(s):  
Sayyed Mahdi Sajedi ◽  
Parviz Ghadimi ◽  
Mohammad Sheikholeslami ◽  
Mohammad A Ghassemi
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Scale Model ◽  
Test Cases ◽  
Planing Craft ◽  
Calm Water ◽  
Space Discretization ◽  
Good Agreement

This paper presents experimental and numerical investigation of stability and rooster tail of a mono-hull high-speed planing craft with a constant deadrise angle. Initially, a one-fifth scale model was tested in a towing tank, which showed porpoising phenomenon at 8 m/s (equal to the speed of sailing). Subsequently, two wedges of 5 and 10 mm heights, based on the boundary layer calculations, were mounted on the aft section of the planing hull. These wedges were shown to increase the lift at the aft section. These experiments were carried out at different speeds up to 10 m/s in calm water. The experimental results indicated that the installed wedges reduced the trim, drag, and the elapsed time for reaching the hump peak, and also eliminated the porpoising condition. All these test cases were also numerically simulated using Star CCM+ software. The free surface was modeled using the volume of fluid scheme in three-dimensional space. The examined planing craft had two degrees of freedom, and overset mesh technique was used for space discretization. The obtained numerical results were compared with experimental data and good agreement was displayed in the presented comparisons. Ultimately, the effect of the wedge on the rooster tail behind the planing craft was studied. The results of this investigation showed that by decreasing the trim at a constant speed, the height of the generated wake profile (rooster tail) behind the craft decreases, albeit its length increases.

Equilibrium and Associated Load Distribution in Ball and Roller Bearings Loaded in Five Degrees of Freedom While Neglecting Friction—Part I: General Theory and Application to Ball Bearings

1989 ◽  
Vol 111 (1) ◽  
pp. 142-148 ◽  
Author(s):  
J. M. de Mul ◽  
J. M. Vree ◽  
D. A. Maas
Keyword(s):  
General Theory ◽  
Load Distribution ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Contact Analysis ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Roller Bearings ◽  
Rolling Element ◽  
Bearing Stiffness

A new, general and consistent mathematical model of highly modular character is presented for calculation of the equilibrium and associated load distribution in rolling element bearings. The bearings may be loaded and displaced in five degrees of freedom. High speed rolling element loading is considered, internal friction is neglected, the material is assumed linearly elastic and the bearing rings are modelled as rigid except for local contact deformation. Either classical Hertzian contact analysis or modern non-Hertzian contact analysis of sophisticated or approximate character is used as applicable. The bearing stiffness matrix is computed analytically and used internally in the iterative bearing equilibrium calculation; its final values may be used for other purposes such as (rotor) dynamics analysis. In Part I, the general theory and application to ball bearings is presented. In Part II, application of the general theory to roller bearings and an experimental verification are presented.

Detection of Cracks in Mistuned Bladed Disks Using Reduced Order Models and Vibration Data

2011 ◽  
Author(s):  
Chulwoo Jung ◽  
Akira Saito ◽  
Bogdan I. Epureanu
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Three Dimensional ◽  
Cost Savings ◽  
Forced Response ◽  
Response Analysis ◽  
Reduced Order Models ◽  
Reduced Order

A novel methodology to detect the presence of a crack and to predict the nonlinear forced response of mistuned turbine engine rotors with a cracked blade and mistuning is developed. The combined effects of the crack and mistuning are modeled. First, a hybrid-interface method based on component mode synthesis is employed to develop reduced order models (ROMs) of the tuned system with a cracked blade. Constraint modes are added to model the displacements due to the intermittent contact between the crack surfaces. The degrees of freedom (DOFs) on the crack surfaces are retained as active DOFs so that the physical forces due to the contact/interaction (in the three-dimensional space) can be accurately modeled. Next, the presence of mistuning in the tuned system with a cracked blade is modeled. Component mode mistuning is used to account for mistuning present in the un-cracked blades while the cracked blade is considered as a reference (with no mistuning). Next, the resulting (reducedorder) nonlinear equations of motion are solved by applying an alternating frequency/time-domain method. Using these efficient ROMs in a forced response analysis, it is found that the new modeling approach provides significant computational cost savings, while ensuring good accuracy relative to full-order finite element analyses. Furthermore, the effects of the cracked blade on the mistuned system are investigated, and used to detect statistically the presence of a crack and to identify which blade of a full bladed disk is cracked. In particular, it is shown that cracks can be distinguished from mistuning.

Nonlinear Excitation Model of Ball Bearing Waviness in a Rigid Rotor Supported by Two or More Ball Bearings Considering Five Degrees of Freedom

2001 ◽  
Vol 124 (1) ◽  
pp. 82-90 ◽  
Author(s):  
G. H. Jang ◽  
S. W. Jeong
Keyword(s):  
Degrees Of Freedom ◽  
Ball Bearing ◽  
Angular Displacement ◽  
Equations Of Motion ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Rigid Rotor ◽  
Nonlinear Load ◽  
Rolling Elements ◽  
Nonlinear Equations Of Motion

This research presents a nonlinear model to analyze the ball bearing vibration due to the waviness in a rigid rotor supported by two or more ball bearings. The waviness of a ball and each races is modeled by the superposition of sinusoidal function, and the position vectors of inner and outer groove radius center are defined with respect to the mass center of the rotor in order to consider five degrees of freedom of a general rotor-bearing system. The waviness of a ball bearing is introduced to these position vectors to use the Hertzian contact theory in order to calculate the elastic deflection and nonlinear contact force resulting from the waviness while the rotor has translational and angular motion. They can be determined by solving the nonlinear equations of motion with five degrees of freedom by using the Runge-Kutta-Fehlberg algorithm. Numerical results of this research are validated with those of prior researchers. The proposed model can calculate the translational displacement as well as the angular displacement of the rotor supported by two or more ball bearings with waviness. It also characterizes the vibration frequencies resulting from the various kinds of waviness in rolling elements, the harmonic frequencies resulting from the nonlinear load-deflection characteristics of ball bearing, and the sideband frequencies resulting from nonlinearity of the waviness interaction.

scholarly journals Soft phototactic swimmer based on self-sustained hydrogel oscillator

2019 ◽  
Vol 4 (33) ◽  
pp. eaax7112 ◽  
Author(s):  
Yusen Zhao ◽  
Chen Xuan ◽  
Xiaoshi Qian ◽  
Yousif Alsaid ◽  
Mutian Hua ◽  
...  
Keyword(s):  
High Speed ◽  
Degrees Of Freedom ◽  
Modular Design ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Ambient Conditions ◽  
Solar Sails ◽  
Living Organisms ◽  
Light Material ◽  
Three Dimensional Space

Oscillations are widely found in living organisms to generate propulsion-based locomotion often driven by constant ambient conditions, such as phototactic movements. Such environment-powered and environment-directed locomotions may advance fully autonomous remotely steered robots. However, most man-made oscillations require nonconstant energy input and cannot perform environment-dictated movement. Here, we report a self-sustained soft oscillator that exhibits perpetual and untethered locomotion as a phototactic soft swimming robot, remotely fueled and steered by constant visible light. This particular out-of-equilibrium actuation arises from a self-shadowing–enabled negative feedback loop inherent in the dynamic light–material interactions, promoted by the fast and substantial volume change of the photoresponsive hydrogel. Our analytical model and governing equation unveil the oscillation mechanism and design principle with key parameters identified to tune the dynamics. On this autonomous oscillator platform, we establish a broadly applicable principle for converting a continuous input into a discontinuous output. The modular design can be customized to accommodate various forms of input energy and to generate diverse oscillatory behaviors. The hydrogel oscillator showcases agile life-like omnidirectional motion in the entire three-dimensional space with near-infinite degrees of freedom. The large force generated by the powerful and long-lasting oscillation can sufficiently overcome water damping and effectively self-propel away from a light source. Such a hydrogel oscillator–based all-soft swimming robot, named OsciBot, demonstrated high-speed and controllable phototactic locomotion. This autonomous robot is battery free, deployable, scalable, and integratable. Artificial phototaxis opens broad opportunities in maneuverable marine automated systems, miniaturized transportation, and solar sails.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

Investigation of Train Dynamics in Passing Through Curves Using a Full Model

Joint Rail ◽  
2004 ◽  
Author(s):  
Mohammad Durali ◽  
Mohammad Mehdi Jalili Bahabadi
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Full Model ◽  
Bogie Frame ◽  
Nonlinear Characteristics ◽  
Train Derailment ◽  
Train Dynamics

In this article a train model is developed for studying train derailment in passing through bends. The model is three dimensional, nonlinear, and considers 43 degrees of freedom for each wagon. All nonlinear characteristics of suspension elements as well as flexibilities of wagon body and bogie frame, and the effect of coupler forces are included in the model. The equations of motion for the train are solved numerically for different train conditions. A neural network was constructed as an element in solution loop for determination of wheel-rail contact geometry. Derailment factor was calculated for each case. The results are presented and show the major role of coupler forces on possible train derailment.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

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