Perturbation Analysis of Texture Effects on Lubricated Devices

2005 ◽  
Author(s):  
Gustavo C. Buscaglia ◽  
Mohammed Jai ◽  
Sorin Ciuperca
Keyword(s):  
Friction Coefficient ◽  
Friction Force ◽  
Perturbation Analysis ◽  
Mathematical Analysis ◽  
Load Capacity ◽  
Second Order ◽  
Numerical Investigations ◽  
First Order ◽  
The Mean

Given a bearing of some specified shape, what is the effect of texturing its surfaces uniformly? Experimental and numerical investigations on this question have recently been pursued, which we complement here with a mathematical analysis. Assuming the texture length to be much smaller than the bearing’s length, we combine homogenization techniques with perturbation analysis. This allows us to consider arbitrary, 2D texture shapes. The results show that both the load capacity and the friction force depend, to first order in the amplitude, just on the mean depth/height of the texture. The dependence of the friction coefficient is thus of second order.

The Effect of Periodic Textures on the Static Characteristics of Thrust Bearings

2005 ◽  
Vol 127 (4) ◽  
pp. 899-902 ◽  
Author(s):  
Gustavo C. Buscaglia ◽  
Ionel Ciuperca ◽  
Mohammed Jai
Keyword(s):  
Friction Coefficient ◽  
Perturbation Analysis ◽  
Mathematical Analysis ◽  
Small Amplitude ◽  
Reynolds Equation ◽  
Static Characteristics ◽  
Thrust Bearings ◽  
Numerical Investigations ◽  
Macroscopic Shape

Given a bearing of some specified macroscopic shape, what is the effect of texturing its surfaces uniformly? Experimental and numerical investigations on this question have recently been pursued, which we complement here with a mathematical analysis based on a seemingly novel combination of homogenization techniques and perturbation analysis. The flow is assumed governed by the Reynolds equation, with cavitation effects disregarded, and the texture length is assumed much smaller than the bearing’s length. The results, which hold true for small-amplitude periodic textures and in the limit of vanishing period, can be summarized as follows: (a) The texture that maximizes the load for a given minimum clearance is no texture at all (i.e., the untextured shape); and (b) the texture that minimizes the friction coefficient is again the untextured shape.

The Second-Order Master Equation

2019 ◽  
pp. 128-158
Author(s):  
Pierre Cardaliaguet ◽  
François Delarue ◽  
Jean-Michel Lasry ◽  
Pierre-Louis Lions
Keyword(s):  
Master Equation ◽  
Mean Field ◽  
Second Order ◽  
Well Posedness ◽  
Mean Field Game ◽  
First Order ◽  
Common Noise ◽  
System P ◽  
The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

A Perturbation Analysis of the Wavemaking of a Ship, with an Interpretation of Guilloton’s Method

1975 ◽  
Vol 19 (03) ◽  
pp. 140-148
Author(s):  
F. Noblesse
Keyword(s):  
Approximate Solution ◽  
Free Surface ◽  
Perturbation Analysis ◽  
Order Approximation ◽  
Second Order ◽  
Regular Perturbation ◽  
First Order ◽  
Perturbation Problem ◽  
Sinkage And Trim ◽  
Basic Ideas

A thin-ship perturbation analysis, suggested by Guilloton's basic ideas, is presented. The analysis may be regarded as an application of Lighthill's method of strained coordinates to a regular perturbation problem. An inconsistent second-order approximation in which the Laplace equation is satisfied to first order, and the boundary conditions both at the free surface and on the ship hull are satisfied to second order, is derived. When sinkage and trim, incorporated into the present analysis, are ignored, this approximate solution is shown to be essentially equivalent to the method of Guilloton.

Friction Characteristics and Adhesion Force Under Low Normal Load

1995 ◽  
Vol 117 (4) ◽  
pp. 569-574 ◽  
Author(s):  
Yasuhisa Ando ◽  
Yuichi Ishikawa ◽  
Tokio Kitahara
Keyword(s):  
Friction Coefficient ◽  
Friction Force ◽  
Adhesion Force ◽  
Normal Load ◽  
Adhesion Forces ◽  
Test Pieces ◽  
Before And After ◽  
Constant Normal Load ◽  
The Mean ◽  
Steel Balls

The friction coefficient and adhesion force between steel balls and flat test pieces were measured during friction under low normal load in order to examine the tribological characteristics. First, the friction coefficients were measured under a constant normal load of 0.8 to 2350 μN, and the adhesion forces were measured before and after each friction. The result showed that the friction coefficient was highest at low normal loads, while the friction force divided by the sum of the normal load and the mean adhesion force was almost constant over the whole range of loads. Second, when the normal load was reduced gradually during friction, friction still acted when the normal load became negative and a pulling off force was applied to the surface. Thus an adhesion force acts during friction and this adhesion force affects the friction force in the same way as the normal load.

Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

Nonlinear dynamic response analysis on gear-rotor-bearing transmission system

2016 ◽  
Vol 24 (9) ◽  
pp. 1632-1651 ◽  
Author(s):  
Shihua Zhou ◽  
Guiqiu Song ◽  
Mengnan Sun ◽  
Zhaohui Ren
Keyword(s):  
Friction Coefficient ◽  
Friction Force ◽  
Nonlinear Dynamic ◽  
Nonlinear Differential Equations ◽  
Transmission System ◽  
Dynamic Responses ◽  
Response Analysis ◽  
Light Load ◽  
Rotor Bearing ◽  
The Mean

A coupled lateral-torsional nonlinear dynamic model with 16-degree-of-freedom (16-DOF) of gear-rotor-bearing transmission system (GRBTS) is developed after comprehensive considering the nonlinear features associated with time-varying meshing stiffness, backlash, transmission error, friction force, input/output load, gravity and gear eccentricity. Based on the nonlinear differential equations, the coupled multi-body dynamic responses of the GRBTS are demonstrated using the Runge-Kutta numerical method, and the effects of friction coefficient and mean load on the dynamic characteristics are investigated. The results show that the friction force could enlarge the vibration amplitude and affect the low frequency components seriously. The mean load excitation has a complicated influence on the coupled GRBTS, and the torsional vibration is the dominate response. Whereas the mean load excitation has a certain extent vibration suppression, and light load and heavy load could no longer effectively control the nonlinear vibration of the GRBTS. With the increasing of rotational speed, the friction coefficient and mean load ranges of the chaotic behavior widen and the chaotic characteristics strengthens. It is shown that small parameter random perturbation might be propagated in the vibration system and lead to relatively large vibration of the system. The contribution to provide a reference for the design and study of gear system.

Effect of Velocity-Slip Boundary Conditions on Jeffery–Hamel Flow Solutions

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
M. A. Al-Nimr ◽  
Vladimir A. Hammoudeh ◽  
M. A. Hamdan
Keyword(s):  
Friction Coefficient ◽  
Skin Friction ◽  
Flow Direction ◽  
Velocity Slip ◽  
Skin Friction Coefficient ◽  
Second Order ◽  
Slip Model ◽  
Wall Angle ◽  
First Order ◽  
Slip Boundary

In the present work, the Jeffery–Hamel flow problem has been studied using both first- and second-order velocity-slip models, and then compared with the no-slip model. The objectives are to observe the behavior of the flow predicted by the two slip models and to establish criteria for using the two velocity-slip models. The study concentrates on examining the effect of the change in the Knudsen number (Kn) on the velocity profiles, magnitude of slip at the wall, and skin friction coefficient. Assuming that a difference between the two slip models of the order of 10% or less justifies the use of the simple first-order model, the transitional Kn numbers have been found. These Kn numbers depend on the flow direction, being either inflow or outflow. Also, there are three distinct regions that specify where to use each of the no-slip, first-order, and second-order slip models. Further, the reversal of the flow has been investigated as a function of the Kn number and for different Re⋅α, where Re is Reynolds number and α is the wall angle. Using the second-order slip models, it is found that as the Kn number increases, reversal occurs at Re⋅α smaller than the 10.31 value at which flow reversal happens in the no-slip model, and increasing the Kn number leads to a reduction in the skin friction coefficient in all cases except when reversal occurs.

A Stochastic Model for Primary Settlers

1970 ◽  
Vol 5 (1) ◽  
pp. 129-170
Author(s):  
D.P. Singh ◽  
A.W. Bryson ◽  
P.L. Silveston
Keyword(s):  
Stochastic Model ◽  
Stochastic Models ◽  
Suspended Solids ◽  
Sewage Treatment ◽  
Sewage Treatment Plant ◽  
Treatment Plant ◽  
Second Order ◽  
Order Model ◽  
First Order ◽  
The Mean

Abstract Stochastic models of process units are useful where the flow and concentrations in a feed stream vary appreciably over long time periods in a random way. Models yield not only the mean, but provide a measure of the variation around the mean. Assuming sedimentation can be described by a rate equation, stochastic models are developed for zero, first and second order rate processes. The zero order model can be rejected because it cannot be made to fit plant data, while the second order model was not developed further because of its complexity. The rate parameter for the first order model was evaluated from 1968 suspended solids data for the Kitchener Sewage Treatment Plant and found to have zero variance. Testing the model against 1966 and 1967 data and shorter period for 1968 showed that the model predicted suspended solids and BOD removals differing on the average from plant results by 10%. The first order stochastic model gives, thus, a satisfactory representation of primary settler performance.

A Pedestal Blocks the Perception of Non-Fourier Motion

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 17-17 ◽  
Author(s):  
O I Ukkonen ◽  
A M Derrington
Keyword(s):  
Modulation Depth ◽  
Discrimination Performance ◽  
Spatial Variations ◽  
Second Order ◽  
First Order ◽  
Direction Discrimination ◽  
Forced Choice Task ◽  
The Mean ◽  
Texture Contrast ◽  
Moving Pattern

We wanted to know whether the mechanisms that discriminate the motion of first-order patterns (defined by spatial variations of luminance) differ from those that detect the motion of non-Fourier or second-order patterns (defined by spatial variations of contrast). To address this question we tested whether motion discrimination performance of first-order and second-order patterns was affected by a pedestal (Lu and Sperling, 1995 Vision Research35 2697 – 2722). A pedestal is a static replica of a moving pattern. We used pedestals with contrast or modulation depth twice the value at which it becomes possible to discriminate the direction of a moving pattern. A two-interval forced-choice task was used to determine how direction discrimination varies with contrast of sine gratings (1 cycle deg−1) and modulation depth of amplitude-modulated gratings presented either alone or with a pedestal. The amplitude-modulated gratings had a 5 cycles deg−1 carrier modulated at 1 cycle deg−1. Three different temporal frequencies (1, 3, and 12 Hz) were studied. Performance with sine gratings was unaffected by the pedestal at all temporal frequencies tested. For amplitude-modulated gratings the pedestal raised the modulation depth at which it became possible to discriminate the direction of motion. This elevation in threshold decreased when the mean contrast of the pattern was high. This result shows that immunity to pedestals of texture-contrast patterns (Lu and Sperling, 1996 Journal of the Optical Society of America13 2305 – 2318) does not generalise to other non-Fourier motion stimuli.

Asymmetric Masking: Luminance Gratings Mask Second-Order Gratings, but Not Vice Versa

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 345-345
Author(s):  
A J Schofield ◽  
M A Georgeson
Keyword(s):  
Modulation Depth ◽  
Broad Band ◽  
Gain Control ◽  
High Amplitude ◽  
Second Order ◽  
Human Vision ◽  
First Order ◽  
Contrast Gain ◽  
Spatiotemporal Information ◽  
The Mean

Human vision can detect spatiotemporal information conveyed by first-order modulations of luminance and by second-order, non-Fourier modulations of image contrast. Models for second-order motion have suggested two filtering stages separated by a rectifying nonlinearity. We explore here the encoding of stationary first-order and second-order gratings, and their interaction. Stimuli consisted of 2-D broad-band static visual noise sinusoidally modulated in luminance (first-order, LM) or contrast (second-order, CM). Modulation thresholds were measured in a two-interval forced-choice staircase procedure. With increasing noise contrast, first-order sensitivity decreased (owing to masking) but sensitivity to contrast modulation increased. Weak background gratings present in both intervals produced order-specific facilitation: LM background facilitated LM detection (the ‘dipper function’) and CM facilitated CM detection. LM did not facilitate CM, nor vice versa, and this is strong evidence that LM and CM are detected via different mechanisms. Nevertheless, suprathreshold LM gratings masked CM detection, but not vice versa. High-amplitude CM masks had little or no effect on CM or LM detection. A broadly tuned divisive gain-control mechanism applied to the first-order filtering stage has been proposed by Foley (1994 Journal of the Optical Society of America A11 1710 – 1719) to account for masking of luminance gratings, and this might also explain the masking of second-order by first-order stimuli. First-order maskers would drive down the effective contrast of the carrier, thus reducing second-order sensitivity. But for second-order maskers the mean contrast, and hence contrast gain, remained constant, independent of modulation depth. Thus second-order gratings would produce no masking effects, as observed.

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