The velocities on the right-hand side of this equation are in Fig. 5-2. If the bearing is stationary and the shaft rotates, the fluid film boundary conditions on the bearing surface are U 1 ¼ 0; V 1 ¼ 0 ð15-2Þ Under dynamic conditions, the velocity on the journal surface is a vector summation of the velocity of the journal center, O 1 , and the journal surface velocity relative to that center. The journal center has radial and tangential velocity components, as shown in Fig. 15-1. The components are de= dt ¼ Cðde=dtÞ in the radial direction and eðdf=dtÞ in the tangential direction. Summation of the velocity components of O 1 with that of the journal surface relative to O 1 results in the following components, U 2 and V 2 (see Fig. 5-2 for the direction of the components): U 2 ¼ oR þ de dt sin y À e df dt cos y ð15-3Þ V 2 ¼ oR dh dx þ de dt cos y þ e df dt sin y ð15-4Þ Here, h is the fluid film thickness around a journal bearing, given by the equation h ¼ C 1 þe cos yðÞ ð15-5Þ After substitution of U 2 and V 2 as well as U 1 and V 1 in the right-hand side of the Reynolds equation, Eq. (15-1), the pressure distribution can be derived in a similar way to that of a steady short bearing. The load components are obtained by integrating the pressure in the converging clearance only ð0 < y < pÞ as follows: W x ¼À2R ð p 0 ð L=2 0 p cos y dy dz ð15-6Þ W y ¼ 2R ð p 0 ð L=2 0 p sin y dy dz ð15-7Þ Converting into dimensionless terms, the equations for the two load capacity components (in the X and Y directions as shown in Fig. 15-1) become W x ¼À0:5e J 12 jUjþ e _ ffJ 12 þ _ ee J 22 ð15-8Þ W y ¼þ0:5e J 11 U Àe _ ffJ 11 À _ ee J 12 ð15-9Þ Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. Here, the dimensionless load capacity and velocity are W ¼ C 2 mU 0 L 3 W and U ¼ U U 0 ð15-10Þ where U ¼ oR is the time-variable velocity of the journal surface and U 0 is a reference constant velocity used for normalizing the velocity. The integrals J ij and their solutions are given in Eq. (7-13). Under steady conditions, the external force, F, is equal to the bearing load capacity, W . However, under dynamic conditions, the resultant vector of the two forces accelerates the journal mass according to Newton’s second law: ~ FF À ~ WW ¼ m ~ aa ð15-11Þ Here, m is the mass of the journal, ~ aa is the acceleration vector of the journal center, ~ FF is the external load, and ~ WW is the hydrodynamic load capacity. Under general dynamic conditions, ~ FF and ~ WW are not necessarily in the same direction, and both can be a function of time. In order to convert Eq. (15-11) to dimensionless terms, the following dimensionless variables are defined: m ¼ C 3 U 0 mL 3 R 2 m; F ¼ C 2 mU 0 L 3 F ð15-12Þ Dividing Eq. (15-11) into two components in the directions of F x and F y (along X and Y but in opposite directions) and substituting the acceleration components in the radial and tangential directions in polar coordinates, the following two equations are obtained: F x À W x ¼ m € ee À me _ ff 2 ð15-13Þ F y À W y ¼Àme € ff À 2m _ ee _ ff ð15-14Þ The minus signs in Eq. 15-14 are minus because F y is in the opposite direction to the acceleration. Here, the dimensionless time is defined as t ¼ ot, and the dimensionless time derivatives are _ ee ¼ 1 o de dt ; _ ff ¼ 1 o d _ ff dt ð15-15Þ Substituting the values of the components of the load capacity, W x and W y , from Eqs. (15-8) and (15-9) into Eqs. (15-13) and (15-14), the following two differential equations for the journal center motion are obtained: FðtÞcosðf À pÞ¼À0:5e J 12 jUjþJ 12 e _ ff þ J 22 _ ee þ m € ee À me _ ff 2 ð15-16Þ FðtÞsinðf À pÞ¼0:5e J 11 U ÀJ 11 e _ ff À J 12 _ ee À me € ff À 2m _ ee _ ff ð15-17Þ Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. Here, FðtÞ is a time-dependent dimensionless force acting on the bearing. The force (magnitude and direction) is a function of time. In the two equations, e is the eccentricity ratio, f is the attitude angle, and m is dimensionless mass, defined by Eq. (15-12). The definition of the integrals J ij and their solution are in Chapter 7. Equations (15-16) and (15-17) are two differential equations required for the solutions of the two time-dependent functions e and f. The variables e and f represent the motion of the shaft center, O 1 , with time, in polar coordinates. The solution of the two equations as a function of time is finally presented as a plot of the trajectory of the journal center. If there are steady-state oscillations, such as sinusoidal force, after the initial transient, the trajectory becomes a closed locus that repeats itself each load cycle. A repeated trajectory is referred to as a journal center locus. 15.3 JOURNAL CENTER TRAJECTORY The integration of Eqs. (15-16) and (15-17) is performed by finite differences with the aid of a computer program. Later, a computer graphics program is used to plot the journal center motion. The plot of the time variables e and f, in polar coordinates, represents the trajectory of the journal center motion relative to the bearing. The eccentricity ratio e is a radial coordinate and f is an angular coordinate. Under harmonic conditions, such as sinusoidal load, the trajectory is a closed loop, referred to as a locus. Under harmonic oscillations of the load, there is initially a transient trajectory; and after a short time, a steady state is reached where the locus repeats itself during each cycle. In heavily loaded bearings, the locus can approach the circle e ¼ 1, where there is a contact between the journal surface and the sleeve. The results allow comparison of various bearing designs. The design that results in a locus with a lower value of maximum eccentricity ratio e is preferable, because it would resist more effectively any unexpected dynamic disturbances. 15.4 SOLUTION OF JOURNAL MOTION BY FINITE-DIF FERENCE METHOD Equations (15-16) and (15-17) are the two differential equations that are solved for the function of e versus f. The two equations contain first- and second-order time derivatives and can be solved by a finite-difference procedure. The equations are not linear because the acceleration terms contain second-power time deriva- tives. Similar equations are widely used in dynamics and control, and commercial Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. software is available for numerical solution. However, the reader will find it beneficial to solve the equations by himself or herself, using a computer and any programming language that he or she prefers. The following is a demonstration of a solution by a simple finite-difference method. The principle of the finite-difference solution method is the replacement of the time derivatives by the following finite-difference equations (for simplifying the finite difference procedure, " FF, " mm and " tt are renamed F, m and t): _ ff n ¼ f nþ1 À f nÀ1 2 Dt ; _ ee n ¼ e nþ1 À e nÀ1 2 Dt ð15-18Þ and the second time derivatives are € ff n ¼ f nþ1 À 2f n þ f nÀ1 Dt 2 ; € ee n ¼ e nþ1 À 2e n þ e nÀ1 Dt 2 ð15-19Þ For the nonlinear terms (the last term in the two equations), the equation can be linearized by using the following backward difference equations: _ ff n ¼ f n À f nÀ1 Dt ð15-20Þ By substituting the foregoing finite-element terms for the time-derivative terms, the two unknowns e nþ1 and f nþ1 can be solved as two unknowns in two regular linear equations. After substitution, the differential equations become F x þ 1 2 e n J 12 ¼ e n J 12 f nþ1 À f nÀ1 2 Dt  þ J 22 e nþ1 À e nÀ1 2 Dt  þ m e nþ1 À 2e n þ e nÀ1 Dt 2  À me n f n À f nÀ1 Dt  2 ð15-21Þ F y À 1 2 e n J 11 ¼Àe n J 11 f nþ1 À f nÀ1 2 Dt  À J 12 e nþ1 À e nÀ1 2 Dt  À me n f nþ1 À 2f n þ f nÀ1 Dt 2  À 2m e nþ1 À e nÀ1 2Dt   f n À f nÀ1 Dt  ð15-22Þ Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. Here, F x and F y are the external load components in the X and Y directions, respectively. Under dynamic conditions, the load components vary with time: F x ¼ FðtÞðcos f À pÞð15-23Þ F y ¼ FðtÞðsin f À pÞð15-24Þ Equations (15-21) and (15-22) can be rearranged as two linear equations in terms of e nþ1 and f nþ1 as follows: Rearranging Eq. (15-21): A ¼ Be nþ1 þ Cf nþ1 ð15-25Þ Rearranging Eq. (15-22): P ¼ Re nþ1 þ Qf nþ1 ð15-26Þ In the following equations, F and m are dimensionless terms (the bar is omitted for simplification). The values of the coefficients of the unknown variables [in Eqs. (15-25) and (15-26)] are A ¼ F X þ e n J 12 2 þ e n J 12 f nÀ1 2 Dt þ J 22 e nÀ1 2 Dt þ 2me n Dt 2 À me nÀ1 Dt 2 þ me n f n À f nÀ1 Dt  2 ð15-27Þ B ¼ J 22 2 Dt þ m Dt 2 ð15-28Þ C ¼ e n J 12 2 Dt ð15-29Þ P ¼ F y À e n J 11 2 À e n J 11 f nÀ1 2 Dt À J 12 e nÀ1 2 Dt À 2me n f n Dt 2 þ me n f nÀ1 Dt 2 À me nÀ1 f n Dt 2 þ me nÀ1 f nÀ1 Dt 2 ð15-30Þ Q ¼À e n J 11 2 Dt À me n Dt 2 ð15-31Þ R ¼À J 12 2 Dt À m Dt 2 f n þ mf nÀ1 Dt 2 ð15-32Þ The numerical solution of the two equations for the two unknowns becomes e nþ1 ¼ AQ ÀPC BQ ÀRC ð15-33Þ f nþ1 ¼ AR ÀPB CR ÀQB ð15-34Þ Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. The last two equations make it possible to march from the initial conditions and find e nþ1 and f nþ1 from any previous values, in dimensionless time intervals of D " tt ¼ oDt. For a steady-state solution such as periodic load, the first two initial values of e and f can be selected arbitrarily. The integration of the equations must be conducted over sufficient cycles until the initial transient solution decays and a periodic steady-state solution is reached, i.e., when the periodic e and f will repeat at each cycle. The following example is a solution for the locus of a short hydrodynamic bearing loaded by a sinusoidal force that is superimposed on a constant vertical load. The example compares the locus of a Newtonian and a viscoelastic fluid. The load is according to the equation FðtÞ¼800 þ800 sin 2ot ð15-35Þ In this equation, o is the journal angular speed. This means that the frequency of the oscillating load is twice that of the journal rotation. The direction of the load is constant, but its magnitude is a sinusoidal function. The dimensionless load is according to the definition in Eq. (15-12). The dimensionless mass is m ¼100 and the journal velocity is constant. The resulting steady-state locus is shown in Fig. 15-2 by the full line for a Newtonian fluid. The dotted line is for a viscoelastic lubricant under identical FIG. 15-2 Locus of the journal center for the load F t ¼ 800 þ 800 sin 2ot and journal mass m ¼ 100. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. conditions (see Chapt. 19). The viscoelastic lubricant is according to the Maxwell model in Chapter 2 [Eq. (2-9)]. The dimensionless viscoelastic parameter G is G ¼ lo ð15-36Þ where l is the relaxation time of the fluid and o is the constant angular speed of the shaft. In this case, the result is dependent on the ratio of the load oscillation frequency, o 1 , and the shaft angular speed, o 1 =o. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. 16 Friction Characteristics 16.1 INTRODUCTION The first friction model was the Coulomb model, which states that the friction coefficient is constant. Recall that the friction coefficient is the ratio f ¼ F f F ð16-1Þ where F f is the friction force in the direction tangential to the sliding contact plane and F is the load in the direction normal to the contact plane. Discussion of the friction coefficient for various material combinations is found in Chapter 11. For many decades, engineers have realized that the simplified Coulomb model of constant friction coefficient is an oversimplification. For example, static friction is usually higher than kinetic friction. This means that for two surfaces under normal load F, the tangential force F f required for the initial breakaway from the rest is higher than that for later maintaining the sliding motion. The static friction force increases after a rest period of contact between the surfaces under load; it is referred to as stiction force (see an example in Sec. 16.3). Subsequent attempts were made to model the friction as two coefficients of static and kinetic friction. Since better friction models have not been available, recent analytical studies still use the model of static and kinetic friction coefficients to analyze friction-induced vibrations and stick-slip friction effects in dynamic Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. systems. However, recent experimental studies have indicated that this model of static and kinetic friction coefficients is not accurate. In fact, a better description of the friction characteristics is that of a continuous function of friction coefficient versus sliding velocity. The friction coefficient of a particular material combination is a function of many factors, including velocity, load, surface finish, and temperature. Never- theless, useful tables of constant static and kinetic friction coefficients for various material combinations are currently included in engineering handbooks. Although it is well known that these values are not completely constant, the tables are still useful to design engineers. Friction coefficient tables are often used to get an idea of the approximate average values of friction coefficients under normal conditions. Stick-slip friction: This friction motion is combined of short consecutive periods of stick and slip motions. This phenomenon can take place whenever there is a low stiffness of the elastic system that supports the stationary or sliding body, combined with a negative slope of friction coefficient, f, versus sliding velocity, U, at low speed. For example, in the linear-motion friction apparatus (Fig. 14-10), the elastic belt of the drive reduces the stiffness of the support of the moving part. In the stick period, the motion is due to elastic displacement of the support (without any relative sliding). This is followed by a short period of relative sliding (slip). These consecutive periods are continually repeated. At the stick period, the motion requires less tangential force for a small elastic displacement than for breakaway of the stiction force. The elastic force increases linearly with the displacement (like a spring), and there is a transition from stick to slip when the elastic force exceeds the stiction force, and vice versa. The system always selects the stick or slip mode of minimum resistance force. In the past, the explanation was based on static friction greater than the kinetic friction. It has been realized, however, that the friction is a function of the velocity, and the current explanation is based on the negative f ÀU slope, see a simulation by Harnoy (1994). 16.2 FRICTION IN HYDRODYNAMIC AND MIXED LUBRICATION Hydrodynamic lubrication theory was discussed in Chapters 4–9. In journal and sliding bearings, the theory indicates that the lubrication film thickness increases with the sliding speed. Full hydrodynamic lubrication occurs when the sliding velocity is above a minimum critical velocity required to generate a full lubrication film having a thickness greater than the size of the surface asperities. In full hydrodynamic lubrication, there is no direct contact between the sliding Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. surfaces, only viscous friction, which is much lower than direct contact friction. In full fluid film lubrication, the viscous friction increases with the sliding speed, because the shear rates and shear stresses of the fluid increase with that speed. Below a certain critical sliding velocity, there is mixed lubrication, where the thickness of the lubrication film is less than the size of the surface asperities. Under load, there is a direct contact between the surfaces, resulting in elastic as well as plastic deformation of the asperities. In the mixed lubrication region, the external load is carried partly by the pressure of the hydrodynamic fluid film and partly by the mechanical elastic reaction of the deformed asperities. The film thickness increases with sliding velocity; therefore as the velocity increases, a larger portion of the load is carried by the fluid film. The result is that the friction decreases with velocity in the mixed region, because the fluid viscous friction is lower than the mechanical friction at the contact between the asperities. The early measurements of friction characteristics have been described by f –U curves of friction coefficient versus sliding velocity by Stribeck (1902) and by McKee and McKee (1929). These f –U curves were measured under steady conditions and are referred to as Stribeck curves. Each point of these curves was measured under steady-state conditions of speed and load. The early experimental f –U curves of lubricated sliding bearings show a nearly constant friction at very low sliding speed (boundary lubrication region). However, for metal bearing materials, our recent experiments in the Bearing and Bearing Lubrication Laboratory at the New Jersey Institute of Technology, as well as experiments by others, indicated a continuous steep downward slope of friction from zero sliding velocity without any distinct friction characteristic for the boundary lubrication region. The recent experiments include friction force measurement by load cell and on-line computer data acquisition. Therefore, better precision is expected than with the early experiments, where each point was measured by a balance scale. An example of an f –U curve is shown in Fig. 16-1. This curve was produced in our laboratory for a short journal bearing with continuous lubrica- tion. The experiment was performed under ‘‘quasi-static’’ conditions; namely, it was conducted for a sinusoidal sliding velocity at very low frequency, so it is equivalent to steady conditions. The curve demonstrates high friction at zero velocity (stiction, or static friction force), a steep negative friction slope at low velocity (boundary and mixed friction region), and a positive slope at higher velocity (hydrodynamic region). There are a few empirical equations to describe this curve at steady conditions. The negative slope of the f –U curve at low velocity is used in the explanation of several friction phenomena. Under certain conditions, the negative slope can cause instability, in the form of stick-slip friction and friction-induced vibrations (Harnoy 1995, 1996). In the boundary and mixed lubrication regions, the viscosity and boundary friction additives in the oil significantly affect the friction characteristics. In Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved. [...]... (due to increasing viscous friction) Although there is a similarity in the shapes of the curves, the breakaway friction coefficient of rolling bearings is much lower than that of sliding bearings, such as journal bearings This is obvious because rolling friction is lower than sliding friction The load and the bearing type affect the friction coefficient For example, cylindrical and tapered rolling elements... conditions, such as in the hip joint during walking Variable friction under unsteady conditions is referred to as dynamic friction Recently, there has been an increasing interest in dynamic friction measurements Dynamic tests, such as oscillating sliding motion, require on-line recording of friction Experiments with an oscillating sliding plane by Bell and Burdekin Copyright 20 03 by Marcel Dekker, Inc All Rights... (Hershey number—see Sec 8. 7.1) 16.2.1 Friction in Rolling-Element Bearings Stribeck measured similar f –U curves (friction coefficient versus rolling speed) for lubricated ball bearings and published these curves for the first time as early as Copyright 20 03 by Marcel Dekker, Inc All Rights Reserved 1902 Rolling-element bearings operating with oil lubrication have a similar curve: an initial negative slope... higher during acceleration than during deceleration For example, the friction coefficient is higher during the start-up of a machine than the friction during stopping This means that the friction is not only a function of the instantaneous sliding velocity, but also a function of velocity history Examples of f ÀU curves under dynamic conditions are included in Chap 17 Copyright 20 03 by Marcel Dekker, Inc... only one aspect in a wider discipline of friction under time-variable conditions Variable friction under unsteady conditions is referred to as dynamic friction There are many applications involving dynamic friction, such as friction between the piston and sleeve in engines where the sliding speed and load periodically vary with time In the last decade, there was an increasing interest in dynamic friction... slope after the breakaway An example is shown in Fig 16 -3 for dry friction of a journal bearing made of a steel shaft on a brass sleeve This curve indicates a considerably higher friction coefficient at the breakaway from zero velocity (about 0.42 in comparison to 0.26 for a lubricated journal bearing half of the breakaway friction) In addition, a dry bearing has a significantly greater gradual reduction... friction coefficient, independent of sliding speed For journal bearings in the hydrodynamic friction region, the friction coefficient f is a function not only of the sliding speed but of the Sommerfeld number Analytical curves of ðR=CÞ f versus Sommerfeld number are presented in the charts of Raimondi and Boyd; see Fig 8 -3 These charts are for partial journal bearings of various arc angles b These charts... simulation is required for design purposes to prevent these vibrations Stick-slip friction is considered a major limitation for high-precision manufacturing In addition to machine tools, stick-slip friction is a major problem in measurement devices and other precision machines A lot of research has been done to eliminate the stick-slip friction, particularly in machine tools Some solutions involve hardware modifications... phase lag was determined empirically A time lag between oscillating friction and velocity in lubricated *This and subsequent sections in this chapter are for advanced studies Copyright 20 03 by Marcel Dekker, Inc All Rights Reserved surfaces was observed and measured earlier It is interesting to note that Rabinowicz (1951) observed a friction lag even in dry contacts The following analysis offers a... coefficient f increases with the roughness because there is increasing interaction between the surface asperities (Rabinovitz, 1965) 16 .3 FRICTION OF PLASTIC AGAINST METAL There is a fundamental difference between dry friction of metals (Fig 16 -3) where the friction goes down with velocity, and dry friction of a metal on soft plastics (Fig 16-4a) where the friction coefficient increases with the sliding velocity . sliding bearings show a nearly constant friction at very low sliding speed (boundary lubrication region). However, for metal bearing materials, our recent experiments in the Bearing and Bearing. been an increasing interest in dynamic friction measurements. Dynamic tests, such as oscillating sliding motion, require on-line recording of friction. Experiments with an oscillating sliding plane. and sleeve in engines where the sliding speed and load periodically vary with time. In the last decade, there was an increasing interest in dynamic friction as well as its modeling. This interest