Vibrational Characteristics of Ball Bearings

1977 ◽  
Vol 99 (2) ◽  
pp. 284-287 ◽  
Author(s):  
P. K. Gupta ◽  
L. W. Winn ◽  
D. F. Wilcock
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Dominant Frequency ◽  
Hertzian Contact ◽  
Analytical Simulation ◽  
Contact Frequency ◽  
Ball Contact ◽  
Vibrational Characteristics ◽  
General Motion

The classical differential equations of motion of the ball mass center in an angular contact thrust loaded ball bearing are integrated with prescribed initial conditions in order to simulate the natural high frequency vibrational characteristics of the general motion. Two distinct frequencies are identified in the analytical simulation and their existence is also confirmed experimentally. One of the frequencies is found to be associated with the Hertzian contact spring at the ball race contact and it is therefore defined as the “elastic contact frequency”, Ωe. The other dominant frequency corresponding to oscillatory motion of the ball in the raceway groove appears to be kinematic in nature and it is, therefore, termed as the “bearing kinematic frequency”, Ωk. It is shown that for a given bearing Ωe and Ωk, vary as, respectively, 1/6 and 1/2 powers of the ball contact load and, therefore, for a given load these frequencies correspond to the natural frequencies of the bearing as applied in any vibrational analysis or simulation.

Motion of a Ball in a Ball Bearing

1973 ◽  
Vol 95 (1) ◽  
pp. 263-268
Author(s):  
H. Portig ◽  
H. G. Rylander
Keyword(s):  
Deformation Theory ◽  
Coulomb Friction ◽  
Ball Bearing ◽  
Digital Simulation ◽  
Equations Of Motion ◽  
Unsteady Motion ◽  
Hertzian Contact ◽  
Contact Deformation ◽  
Normal Forces ◽  
Simulation Language

A method is developed which allows the digital simulation of the unsteady motion of a single ball constrained only by two moving bearing races. Any desired motion of the races can be simulated. Normal forces acting on the ball are calculated by Hertzian contact deformation theory. If there is slippage between ball and races, Coulomb friction is assumed to occur. Solutions to the differential equations of motion were obtained on a computer with the digital simulation language MIMIC. The phenomenon of ball control as well as the behavior of the ball as it reached a controlled state from rest were observed. This analysis can produce more realistic results than methods that assume that the ball is controlled at all times, especially when the races are radially or angularly displaced with respect to each other.

Hertz Contact Force Model With Permanent Indentation in Impact Analysis of Solids

1992 ◽  
Author(s):  
Hamid M. Lankarani ◽  
Parviz E. Nikravesh
Keyword(s):  
Contact Force ◽  
Impact Analysis ◽  
Equations Of Motion ◽  
Solid Particles ◽  
Hertzian Contact ◽  
Contact Period ◽  
Force Model ◽  
Unknown Parameters ◽  
Contact Force Model ◽  
Energy And Momentum

Abstract A continuous analysis method for the direct-central impact of two solid particles is presented. Based on the assumption that local plasticity effects are the sole factor accounting for the dissipation of energy in impact, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are analytically evaluated in terms of a given coefficient of restitution and velocities before impact. The equations of motion of the two solids may then be integrated forward in time knowing the variation of the contact force during the contact period. For Illustration, an impact of two soft metallic particles is studied.

Rolling Bearing Parametric Excitation of a Jeffcott Rotor System

2020 ◽  
Author(s):  
Ghasem Ghannad Tehrani ◽  
Chiara Gastaldi ◽  
Teresa Maria Berruti
Keyword(s):  
Parametric Excitation ◽  
Rotational Speed ◽  
Input Parameter ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Rolling Bearing ◽  
Mean Value ◽  
Hertzian Contact ◽  
Rolling Bearings ◽  
Jeffcott Rotor

Abstract Rolling bearings are still widely used in aeroengines. Whenever rotors are modeled, rolling bearing components are typically modeled using springs. In simpler models, this spring is considered to have a constant mean value. However, the rolling bearing stiffness changes with time due to the positions of the balls with respect to the load on the bearing, thus giving rise to an internal excitation known as Parametric Excitation. Due to this parametric excitation, the rotor-bearings system may become unstable for specific combinations of boundary conditions (e.g. rotational speed) and system characteristics (rotor flexibility etc.). Being able to identify these instability regions at a glance is an important tool for the designer, as it allows to discard since the early design stages those configurations which may lead to catastrophic failures. In this paper, a Jeffcott rotor supported and excited by such rolling bearings is used as a demonstrator. In the first step, the expression for the time–varying stiffness of the bearings is analytically derived by applying the Hertzian Contact Theory. Then, the equations of motion of the complete system are provided. In this study, the Harmonic Balance Method (HBM) is used to as an approximate procedure to draw a stability map, thus dividing the input parameter space, i.e. rotational speed and rotor physical characteristics, into stable and unstable regions.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

scholarly journals Elastic shocks in relativistic rigid rods and balls

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180858 ◽  
Author(s):  
João L. Costa ◽  
José Natário
Keyword(s):  
Free Boundary ◽  
Minkowski Space ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Space Time ◽  
Spherically Symmetric ◽  
Soft Phase ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

We study the free boundary problem for the ‘hard phase’ material introduced by Christodoulou in (Christodoulou 1995 Arch. Ration. Mech. Anal. 130 , 343–400), both for rods in (1 + 1)-dimensional Minkowski space–time and for spherically symmetric balls in (3 + 1)-dimensional Minkowski space–time. Unlike Christodoulou, we do not consider a ‘soft phase’, and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

Resonance Raman spectra of highly reduced iron porphycenes

2006 ◽  
Vol 10 (11) ◽  
pp. 1271-1284 ◽  
Author(s):  
Rabbani M. Gulam ◽  
Saburo Neya ◽  
Junji Teraoka
Keyword(s):  
Raman Spectra ◽  
Anion Radical ◽  
Excited Electronic State ◽  
Resonance Raman ◽  
Vibrational Analysis ◽  
Redox States ◽  
Vis Spectroscopy ◽  
Vibrational Characteristics ◽  
Excitation Laser ◽  
Resonance Raman Spectra

The redox behavior of iron porphycenes using the sodium mirror contact technique is reported. The resonance Raman spectra are obtained for each redox state, in order to explore the vibrational characteristics of these species in different redox states. The observed resonance Raman behavior of Fe II porphycene anion radical and its dianion is interpreted using the vibrational analysis of free-base porphycene anion radical and dianion. For the first time, a species generated in the fourth reduction step, is confirmed by UV-vis spectroscopy and is assigned to Fe I porphycene π dianion. The dependence of resonance enhancements of Raman bands for the Fe I porphycene π dianion on excitation laser reveals structural distortion along the NC a C a N or C b C a C a C b segments of the pyrrole-pyrrole direct connection in the excited electronic state.

Cayley kinematics and the Cayley form of dynamic equations

2005 ◽  
Vol 461 (2055) ◽  
pp. 761-781 ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Euler Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Matrix Form ◽  
Dynamic Equations ◽  
Cayley Transform ◽  
Higher Dimensional ◽  
The Matrix ◽  
General Motion

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N –dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N –dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N –dimensional body as pure rotations of an ( N + 1)–dimensional body.

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