Friction (1): 7 39 (014) DOI 10.1007/s40544-013-0034-y ISS 3-7690 REVIEW ARTICLE Theory and contents o rictional mechanics Ping HUAG, Qianqian YAG * School o Mechanical and Automotive Engineering, South China University o Technology, Guangzhou 510640, China Received: 18 September 013 / Revised: 1 ovember 013 / Accepted: 19 ovember 013 The author(s) 013. This article is published with open access at Springerlink.com Abstract: In this paper, we irst discuss the development o the ield o tribology, and highlight some o the main problems encountered in this area, such as lack o systematicness, loose correlation, and inadequate ocus on the microscopic perspective. Then, we provide basic ormulas o rictional mechanics while considering the riction eect on classical mechanics ormulae. In order to carry out the rictional mechanics analysis, we irst classiy the interace. According to the size analysis o surace ilms, the manuacturing roughness o the surace, the contact width, and the roller radius o the rolling contact bearing, rictional mechanics has the eatures o interace mechanics, while interaces are classiied based on the presence or absence o a medium. Based on the classiication, we urther analyze the pressure and rictional stress o sliding and rolling riction problems without a medium, such as a slider, wedge key, and V belt. We also analyze problems with a medium, such as journal and rolling contact bearings. By comparing these results with those o classical mechanics without considering riction, we see that (1) riction causes deviations in the result or classical mechanics which does not consider riction, and () i the rictional stress and normal pressure aect each other, their interaction should be considered simultaneously. Finally, we summarize the riction problems, namely, sliding and rolling, with and without a medium, and deormed and non-deormed. From our analysis, we propose two conclusions. First, the rictional mechanics problem is a deviation o the classical mechanics problem, and secondly, rictional stress and normal pressure inluence each other. Keywords: rictional mechanics; interace; normal pressure; rictional stress 1 Introduction Tribology is an applied subject that involves the study o riction, lubrication, and wear between two relatively moving suraces [1]. Frictional phenomena have been recognized or thousands o years, such as rubbing wood to generate a ire, the invention o wheels, and the use o animal grease or lubrication purposes, etc. [ 5]. In the 15th century, Leonardo Da Vinci began the systematic study o riction, and Coulomb proposed the classical riction laws based on previous studies, that is, the rictional orce is proportional to the normal load [1 5]. Then, the molecular riction theory, electrostatic riction theory, and adhesion theory laid the oundation or modern * Corresponding author: Qianqian YAG. E-mail: qqyangscut@163.com riction theory. Modern lubrication studies began in 1886, when Reynolds derived the hydrodynamic lubrication ormulas based on the indings obtained by Tower [6]. The study o wear commenced much later, and although many wear theories were proposed in the latter hal o the 0th century, none o them have been widely accepted [7, 8]. The combined study o riction, wear, and lubrication began in 1966 when Jost published his report. The report, which estimated that nearly 1/3 1/ o the energy worldwide was wasted because o riction and wear, attracted the attention o many persons, and subsequently, the study o tribology was gradually developed. Because tribology covers such a wide range o topics and there is a theoretical system o tribology, some o the main problems remain unsolved. First, the research contents and methods are dierent [9 1]
8 Friction (1): 7 39 (014) because tribology involves a combination o riction, wear, and lubrication, which all dier in terms o research contents and approach. Friction caused by roughness is closely related to elastic deormation, so that contact mechanics is its main oundation. With respect to wear, a broken material is oten considered, and it is based on material science and racture mechanics. With respect to hydrodynamic lubrication, luid mechanics is used to analyze the ormation and perormance o the lubrication ilm, and some o its research contents and methods are also based on subjects such as chemistry and physical chemistry. Thereore, the dierent research contents and methods relate not only to the characteristics o the three branches, but also to many other subjects. Another eature o tribology problems is their complexity. Because the ormation and working mechanisms o riction, wear, and lubrication are complicated, no single theory can adequately explain these phenomena. Thereore, most tribological problems to be solved are related to solid mechanics, luid mechanics, heat transmission, and chemistry, etc. [13 15]. Surace contact is very important in riction mechanics. Hertz solved the classical contact problem and developed contact mechanics. According to his study, ater the loading o line contacts and point contacts, suraces will deorm to orm a very small contact area. The Hertzian stress σ H is used to determine the stress on the suraces. For contact between two cylinders, the Hertzian stress W, while or H contact between two balls, 3 W. The results H indicate that pressure is not linearly proportional to the load W, as is experienced in common rictional problems involving contact between suraces [16]. Moreover, a macroscopic study o tribology is considered to be not suicient. Generally, the basic approach used to solve tribology problems has been to take a microscopic approach [9 19]. In 193, Bradley provided an adhesive orce calculation equation or two rigid balls to consider the adhesive work between two suraces [0]. However, in his model, the objects were considered to be rigid. Then, Johnson, Kendal, and Roberts presented a stricter theory to analyze the contact between elastic balls, which is known as the JKR adhesive theory [1]. In their theory, a more signiicant result was that they gave a critical negative orce F ad which must be applied to separate the two suraces. In 1975, Derjaguin, Muller, and Toprov put orward an equation known as the DMT adhesive contact model, which can be used to consider the relationship between the contact radius, load, and adhesive orce []. These models are shown in Fig. 1. Fig. 1 Three dierent contact models o ball-plane contact [17]: (a) Rigid contact; (b) Hertz contact or JKR contact; (c) adhesive contact. Although a microscopic analysis is the best way o studying the tribological phenomena and understanding the laws o tribology, the relationship between the subject areas is weak i not also considered rom a macroscopic perspective. Furthermore, because tribological problems involve many unique and random properties, and research into tribology lacks a macro and systematical approach, it has been believed that the establishment o a general theory or tribology is diicult or even impossible. In the present paper, we discuss the extraction o the main inluential actors in order to observe and solve the tribological problems using mechanics, with the aim being to obtain more realistic solutions. In our paper, the relationship between the normal pressure and the rictional stress is used to setup the theoretical system o rictional mechanics. Some amount o micro research achievements must thereore be used to complete such a theoretical system o rictional mechanics. Objects and eatures studied The objects o rictional mechanics are interaces, each o which consists o a rictional pair. The apparent smooth surace actually has many irregular peaks and valleys when viewed with a microscope, and they are ormed during manuacturing processes such as cutting and grinding, because o plastic deormations, vibrations, etc. [3, 4]. The micro roughness and
Friction (1): 7 39 (014) 9 physical chemistry status o the suraces are important actors that aect tribological behavior and processes [17]. Beneath the surace, there is a strengthened layer that is several tens o micrometers thick, going rom the heavy deormed layer to the light deormed layer. Above them, there is an amorphous or microcrystalline layer that is due to melting, lowing, and then rapid cooling, which takes place during manuacturing. Moreover, the oxidizing layer, adsorbed ilm, and/or polluted ilm are on the top. Their common sizes are shown in Fig. in absolute and logarithmic scales. Based on Fig., we classiy the interaces studied in rictional mechanics as those that are with and without a medium, as shown in Fig. 3 [17, 5]. Fig. 3 Interaces without or with medium. 3 Basic equations o rictional mechanics The purpose o rictional mechanics is to develop the relationship between the rictional stress and the normal pressure on the interace, and to solve their distributions using mechanics motions or equilibrium equations and rictional law. 3.1 Mechanics equations While considering the rictional orce, ewton s second law in vector orm can be expressed as ollows: F ma F (1) where the rictional orce F cannot be considered as an active orce, but as a resistance orce induced by the normal load. For dry riction with motion, Coulomb s law can be written as ollows [1]: F () where is the normal load, F is the rictional orce on the contact surace, and is the riction coeicient. Coulomb s law can be deined as the riction coeicient o dry riction, that is, the rictional orce is proportional to the normal load. From a macroscopic perspective, it is an empirical ormula, but in order to use it on a microscopic level to solve the pressure and rictional stress o the interace, we can write a micro- Coulomb law as ollows: p (3) Fig. Comparison o size o surace layers: (a) Common coordinate height; (b) logarithmic coordinate height. where is the rictional stress, is the riction coeicient, and p is the normal pressure. Although Eq. (3) cannot be proven, it is still widely used on a microscopic level.
30 Friction (1): 7 39 (014) For two-dimensional problems, i we set the moving direction as x and the normal direction as y, the equilibrium equations or rictional mechanics will be: along x direction: ( x)dx F along y direction: p( x)dx x1 x x1 x x ( x) 1 to z axis: p( x)( xx )dx Fh x 1 1 micro Coulomb law: ( x) p( x) x where x 1 and x are the two ends o the length o the body in the x direction, p(x) is the normal pressure o the interace, h is the arm o F to Point x 1, and (x x 1 )/ is the arm o the normal load to Point x 1. 3. Elastic deormation equation The elastic deormation o the surace in the line contact is given as the ollowing equation [6]. x e vx ( ) ps ( )ln( sx) dsc πe x 0 where E is the equivalent elastic modulus o the 1 materials o the two contact suraces, E 1 1 1 1, E 1 and E are the elastic modulus E E 1 o the materials o the two contact suraces, respectively, 1 and are the Poisson s ratios o the materials o the two contact suraces, respectively, and c is the integral constant to be determined. For the point contact, the surace elastic deormation can be achieved as ollows [6]: pst (,) vxy (, ) dsdt π E ( xs) ( yt) 3.3 Equations or interace with luid medium While a medium exists in the interace, the problem with the boundary ilm ormed by the medium can be treated as a dry riction problem. Thereore, no additional mechanics equations are needed. However, i there is a luid ilm, Reynolds equation, which is a simple luid mechanics equation, should be used. The two-dimensional Reynolds equation is given as ollows [1]: (4) (5) (6) 3 3 h p h p ( h) 6U x x y y x where is the density o the lubricant, h is the ilm thickness, p is the pressure, is the viscosity o the lubricant, x is the coordinate in tangential direction to the surace velocity U, y is the coordinate vertical to the velocity, and U is the surace velocity. Coulomb s law can no longer be used to obtain the rictional stress between the interace o the solid and luid, but the constitutive equation o a luid such as the ewtonian luid can be used. For a ewtonian luid, the constitutive equation is given by [1]: u z where is the shear stress, z is the coordinate across the ilm thickness, and u is the luid velocity. 4 Sliding rictional mechanics Generally, the rictional orce may not only cause an increase in the shear stress, but it may also change the normal pressure. 4.1 Slider with no medium First, a slider riction problem is used as an example to show the inluence o the rictional stress on the normal pressure. I the slider remains stationary or moves uniormly, the rictional mechanics equations are given by: in x direction: ( x)dx F l l to z axis : p( x) xdx Fh 0, micro Coulomb s law: ( x) p( x) 0 l in y direction: p( x)dx where x and y are the respective coordinates in the tangential and normal directions to the velocity, z is the axis vertical to the paper, F is the tangential load, is the normal load, and l is the length o the slider in the x direction. For convenience, assume that the normal pressure distribution has a linear relationship with the coordinate x: l 0 (7) (8) (9)
Friction (1): 7 39 (014) 31 px ( ) ckx (10) where c and k are the constants to be determined. I they are determined, the normal pressure distribution p(x) and the rictional stress (x) are known. (1) o riction (F = 0) or traction F is in the line o the interace (h = 0), as shown in Fig. 4(a). We have: px ( ) p 0 l ( x) 0 no riction ( x) p h 0 0 l () Small riction l 0 F. The normal pressure 6h and the rictional stress are as shown in Fig. 4(b), and are given as: 6Fh 1Fh px ( ) x 3 l l l 6Fh 1Fh ( x) x 3 l l l 0 x l (3) Large riction l l F. Some parts o the 6h h slider have no normal pressure or rictional stress, as shown in Fig. 4(c). 0 0 xb px ( ) ( x b) p b x l ( l b) 0 0 x b ( x) ( x b) p b x l ( l b) 5l 3hF where b ; and 4 p 6hF 3l (4) Critical situation l F. With the exception h o the ront end o the slider, there is no normal pressure or rictional stress on the other parts o the interace, as shown in Fig. 4(d). (5) Turnover situation l F. The object cannot h move uniormly or be balanced. An application o the results above is the wedge Fig. 4 Frictional stress and pressure distribution when considering riction: (a) o riction; (b) small riction; (c) large riction; (d) critical status. key in the design o mechanical elements [7, 8]. Ater installation, the working suraces o the wedge key are the upper and lower ones, as shown in Fig. 5(a). Without the rotation, there is only a normal load acting on the working suraces, as shown in Fig. 5(b), where p 0 = /lb, l is the length o the key, and b is the width. When rotating with a torque, a rictional orce is produced to resist the torque because there is a relative motion between the shat and the hub [8]. This tendency produces a micro twist deormation so that the pressure changes along the length and width o the key. The resultant o the pressure no longer passes the center o the shat, but is at a distance o x 0, as shown in Fig. 5(c). For convenience, consider the key and the shat as one body, and the resultant orce o the bottom pressure o the cylinder is substituted by. I we assume that the pressure is linearly proportional to the length o the key, it will have a triangular distribution, as shown in Fig. 5(c), where x 0 b/6 and y 0 d/ [7, 8]. We can write the balance equation to the shat center as: T x y d / (11) 0 0 where T is the torque, is the riction coeicient, and d is the diameter o the shat. Then, the resultant orce o the pressure is as ollows: T 6T (1) x y d / b 6 d 0 0 The maximum pressure p max is on the right end o
3 Friction (1): 7 39 (014) Fig. 5 Interace stress analysis o wedge key [7, 8]: (a) loads on wedge key; (b) beore rotation; (c) rotation. the key, which is double the average pressure p 0, and the rictional stress can also be obtained. p T 1T p bl ( x y d/) bl ( b6 d) bl max 0 max p max 0 0 1T ( b 6 d) bl 4. Slider with medium (13) Three types o slider with the medium are shown in Fig. 6 [7, 8]. (a) Two parallel suraces: i we do not consider the hydrostatic situation, the medium can be only the boundary ilm, that is, boundary lubrication. Because the rictional orce is very small, the normal pressure distribution is hardly inluenced by the medium. Thereore, the normal pressure can be considered to be uniorm, as shown in the lower igure o Fig. 6(a). Thereore, the rictional stress o the boundary lubrication obeys Coulomb s law in Eq. (3). Because the pressure is uniorm, the rictional stress o the parallel slider is also uniorm. However, the dierence with the dry riction is that the riction coeicient o the boundary lubrication relates to the strength o the boundary ilm, but it is much smaller than the dry riction coeicient. (b) Divergent wedge: it cannot exist stably in practice, but quickly changes to Case (a) or (c). (c) Convergent wedge: this is the hydrodynamic lubrication status. The pressure distribution can be obtained to solve Reynolds equation, as shown in the lower igure o Fig. 6(c). The expression or the pressure is as ollows [6]: 6U l 1 hh 1 1 1 p kh h h h 1 1 h h h 1 (14) h h where k 1, h is the ilm thickness, h 1 is the ilm h1 thickness o the inlet, h is the ilm thickness o the outlet, and l is the length o the slider. Its rictional stress cannot be obtained rom Coulomb s law in Eq. (3), but can be obtained rom the constitutive equation in Eq. (8). For any point x, the rictional stress on the upper and lower interaces are: h,0 p h v x h (15) Fig. 6 Pressure and rictional stress o slider with inter-medium: (a) Parallel; (b) divergent; (c) convergent. where 0 and h are the shear stress (rictional stress) o the upper and lower suraces, respectively. In Fig. 7, a dimensionless solution o pressure P, ilm thickness H, lower interace rictional stress 0, and upper interace rictional stress h are given or the slider with the medium [6]. It clearly shows that the pressure and rictional stress with the medium are very dierent rom those in the case o dry riction.
Friction (1): 7 39 (014) 33 Reynolds equation in Eq. (16), the pressure distribution can be obtained, as shown in Fig. 8(b) [8]. Compared with the two igures o Fig. 8, it can be seen that the medium signiicantly inluences the interace pressure. The phenomenon where the pressure deviates rom the original signiies that the solution o the hydrodynamic solution is a deviation o the nonhydrodynamic one. Fig. 7 Pressure, ilm thickness, and rictional stresses o up and down interaces. 4.3 Journal bearing (hydrodynamic lubrication) For a journal bearing with hydrodynamic lubrication, the bearing was extended along the circumerence, and Reynolds equation Eq. (6) should be expressed by the cylindrical coordinates [5, 6], that is: 3 3 h p h p Uh 6R y y (16) where is the angular velocity coordinate, R is the radius o the journal bearing, and h is the ilm thickness (the shape o the clearance) [8]: h ecos c c (1 cos ) (17) where e is the eccentricity, c is the clearance, = e/c is the eccentricity ratio, and is the circumerential coordinate starting rom the maximum ilm thickness. I there is no rotation, the pressure is shown as in Fig. 8(a) [8]. While there is a rotation, the hydrodynamic eect must be considered so that a hydrodynamic lubrication ilm will be ormed. By solving 5 Rolling rictional mechanics 5.1 Rolling riction without medium Depending on the actions and riction mechanics, rolling can be divided into our types: (1) Free rolling; () traction pure rolling; (3) torque driven rolling; and (4) sliding-rolling, as shown in Fig. 9. First, we discuss the rictional mechanics o rigid rolling. For rigid rolling, the normal pressure p and the rictional stress tend to ininity because the contact area o the line and point is zero [9]. However, the total normal load and rictional orce have limited values. 5.1.1 Free rolling Free rolling means that there is no driving orce or torque. In other words, there is no relative motion between the moving pair at the contact point. I the Fig. 8 Pressure distribution o journal bearing: (a) Without rotation; (b) with rotation. Fig. 9 Types o rolling: (a) Free rolling; (b) traction rolling; (c) torque driven; (d) sliding-rolling.
34 Friction (1): 7 39 (014) external resistance is not considered, such as air resistance, there is no riction at the contact point, that is, [9]: p A 0 (18) It should be noted that Eq. (18) is only suitable in the ideal case. In reality, the rictional orce is not equal to zero because o the external resistance. Ideal status: It is generally believed that or rolling, there must be some rictional orce at the contact point. However, the rictional orce is a orce that resists the motion or motion tendency, while the contact point is the instant center o motion o the object so that its velocity is equal to zero. Because there is no tendency to move, there is no rictional orce on the contact point. It should be noted that although there is no rictional orce, ree rolling can occur. Furthermore, the absence o a rictional orce means that the surace is smooth. However, because o external resistances, such as air resistance, there will be a rictional orce that balances the resistance. Also, because there is a resistance, it will slow down the rolling o the wheel until the rotational speed becomes zero. 5.1. Traction pure rolling It should also be noted that the pure rolling driven by a traction F may have two rolling types, as shown in Fig. 10. Because the traction is greater or less than the total resistance, the rolling accelerates or decelerates, respectively, and as the traction becomes equals to the total resistance, the rolling speed becomes constant [9]. A traction pure rolling has no relative velocity at the contact point driven by an outside traction to have the rictional pair to roll. From the viewpoint o rictional mechanics, because there is no relative motion, the rictional orce is less than the maximum statics rictional orce. F F F (19) max The rolling riction coeicient is deined as the ratio o the rolling rictional moment to the normal load, that is, FR k e (0) where k is the rolling rictional coeicient. The dimensional rolling riction coeicient k is the length. 5.1.3 Pure rolling driven by torque Pure rolling driven by a torque is the case where there is rolling at the contact point having no relative velocity. Such a motion is similar to two riction wheels rolling with no sliding. Whether a car rides or a person walks on the ground is in this way to move, that is, one wheel is the car s wheel or the person s legs (radius) and eet (wheel rim), and another wheel is the earth. From the perspective o rictional mechanics, i there is no sliding, there is no relative motion at the contact point. Thereore, the riction is a static riction, and the rictional orce at the contact point is smaller than the maximum rictional orce o the contact surace. F F (1) max max where max is the maximum static rictional coeicient. 5.1.4 Sliding-rolling Sliding-rolling can be considered to be a combination o sliding and pure rolling, as shown in Fig. 9(d). When Eq. (1) is not satisied, the wheel appears to slip to the ground surace so that [9]: F F () max Fig. 10 Rolling riction coeicient. Thereore, there is a relative motion at the contact point so that both sliding and rolling exist simultaneously.
Friction (1): 7 39 (014) 35 5. Rolling under elastic deormation The elastic deormation o a cylinder or a ball making contact with a rigid plane can be obtained by Hertzian contact theory [0, 30]. Such an elastic deormation and the corresponding normal pressure are shown in Fig. 11. x p p 1 H b x p p 1 H b (3) where p H is the maximum Hertzian pressure and b is the hal-width o the contact region. Applying the micro Coulomb law in Eq. (3) as the relationship between the rictional stress and the normal pressure, and ater analyzing the sliding o elastic bodies, Buler concluded that i only the two elastic constants (the shear modulus G and Poisson s ratio v) o the contact bodies are the same, the rictional stress does not aect the normal pressure [31]. The analysis made by Johnson [9] shows that i the two materials are dierent, the rictional stress will change the distribution o the normal pressure and the contact width, as shown in Fig. 1. While the riction coeicient = 0.1, the elastic coeicient 1 (1 v ) / G (1 v ) / G 1 1 0., the center (1 v ) / G (1 v ) / G 1 1 x 0 o the normal pressure will move by about 0.1a, and the width o the contact region will increase by about 0.08%. Johnson s conclusion was that because the riction coeicient exceeds 1.0 and the combination coeicient o most materials does not exceed 0.1, the inluence o the rictional stress on the contact pressure distribution and movement o its center are negligible. Fig. 1 Inluence o elastic constant to pressure distribution o sliding cylinder [9]. 5.3 Rolling riction with medium (elastohydrodynamic lubrication) I we do not consider the riction, the contact pressure can be solved by Hertzian contact theory [9 31]. For the line contact, the contact pressure is shown in Fig. 11. While there is a lubricant, the eect o riction must be considered, such as in gears and rolling contact bearings, etc. The pressure can be obtained by solving Reynolds equation in Eq. (7) and the elastic deormation in Eq. (5) or (6) together, which is the EHL solution. The non-dimensionless pressure P, ilm thickness H, and shearing stresses 0 and h are shown in Fig. 13, which is a typical EHL solution or a line contact. The pressure is not the same as the Hertzian stress, but it extends to outside the inlet, and the second pressure peak can be ound at the outlet region, where a corresponding necking also appears [3 35]. Fig. 11 Hertzian stress and deormation in line contact. Fig. 13 Pressure and ilm thickness o EHL on line contact [6].
36 Friction (1): 7 39 (014) 5.4 Transverse elastic rolling-sliding without medium (elastic sliding o belt transmission) The pressure and the rictional stress o the belt transmission are based on the Euler equation or lexible bodies. It can be expressed as ollows [7, 8]: T T e (4) where T is the tension o the contact arc to the angle, T is the tension o the loose side, and is the riction coeicient. With the orce balance conditions, the normal load o the belt transmission is equal to [8]: T e (5) where is the normal load and starts rom the point at which the loose side belt enters the wheel. To derive Eq. (5) or angle, we use the ollowing pressure distribution. p T e R (6) where R is the radius o the wheel and p is the normal pressure. Because the pressure is related to the radius R o the belt wheel, the pressure on the smaller belt wheel R 1 is larger than that on the larger belt wheel. Using Coulomb s law in Eq. (3), we can obtain the ollowing rictional stress rom Eq. (6). T e R where is the rictional stress. (7) Figure 14(a) shows the total stresses on the belt transmission, while the tension distribution according to Eq. (4) and the elastic sliding arcs on the smaller and larger wheels are illustrated in Fig. 14(b) [8]. The analysis o the sliding-rolling o the belt is as ollows [9]. From Eq. (3), it is known that the extension o the belt is dierent because it has a dierent tension, but the wheels have no deormation. Thereore, as the belt passes the belt wheels, it appears to be sliding because the extension varies gradually. This is called elastic sliding. The tension strains o the loose and tension sides have the ollowing relationship. T T (8) 1 1 where is the tension ratio and is the tension strain. While the belt works normally, the elastic sliding only occurs in the area o the contact arcs, where the belt leaves the wheels [8], as shown in Fig. 14(b). This is because the dierences in the velocity o the belt and wheels are the maximum. Thereore, elastic sliding starts rom these two points. By using Eq. (8), the ollowing elastic sliding ratio can be obtained. ( T T ) (9) 1 The contact arc at which there is no elastic sliding is called the static arc. As the transmission power increases, the elastic sliding arc will also increase. As the elastic sliding region extends to the whole contact arc, the total rictional orce reaches a maximum value. Ater that, i the power continues to increase, there will be signiicant sliding between the wheel and the belt, that is, the slip and belt transmission will ail to work. Fig. 14 Force on transmission belt [7, 8]: (a) All stresses; (b) tensions and elastic sliding arcs.
Friction (1): 7 39 (014) 37 6 Discussions Friction is the primary cause o the tangential orce that exists on an interace. Whether it is a sliding, rolling, or sliding-rolling motion, riction will exist on an interace and produce a rictional orce. The rictional orce that exists during movement can usually be determined by measuring the riction coeicient. Although in engineering, it is useul to calculate the rictional orce using the rictional coeicient, the inal tangential orce should still be determined by balancing the external orces. Besides, there is also a rictional orce in a stationary interace, that is, a static rictional orce. A static rictional orce varies rom zero to its maximum value according to the magnitude o an external orce. Thereore, the determination o the tangential orce is also based on balancing the external orces. ow, we list all o the results above in Table 1 to clariy the dierence between cases involving riction and non-riction, deormation and non-deormation, and with and without a lubricant. It can be seen that: (1) There are dierences in the results or cases that both consider and do not consider riction; () The normal pressure and the rictional stress inluence each other; (3) I there is a lubricant, the eect o luid-solid coupling is signiicant. Table 1 Comparison o pressure and rictional stress under dierent working conditions. o riction Friction without medium Friction with medium Sliding Rigid rolling Elastic rolling without medium Elastic rolling with medium Rolling p A 0 p x ph 1 b x p ph 1 b Transverse elastic rolling-sliding without medium T T p e ; e R R
38 Friction (1): 7 39 (014) The rictional orce is usually considered to be in the direction opposite to the moving direction. For the sliding motion, the rictional orce can be easily determined, and it is along the tangent direction o the interace. However, or the rolling motion, the rictional orce may be the same as the direction o motion. Most research has ocused on determining the inluences o dierent actors on the tribology phenomenon on a microscopic level. The establishment o rictional mechanics theory requires the synthesis o more microscopic studies that are combined with results obtained macroscopically. It is believed that this can contribute to the transition o tribology rom an experiential subject to an analytical subject. 7 Conclusions Based on the discussion and analysis above, we presented a set o basic equations o rictional mechanics. Relationships such as Coulomb s law, which exist between the rictional orce and the normal pressure, were added to the general mechanics equations without considering riction. By comparing cases involving sliding and non-sliding and with and without lubricants, the ollowing conclusions were made: (1) The rictional mechanics deviates rom the classical mechanics problem that does not consider riction, that is, the solution o a rictional mechanics problem can be considered to be a variant o the classical mechanics solution. While the rictional orce or riction coeicient tends to zero, the solution becomes the original one. () The rictional orce and the normal load inluence each other, which means that the normal pressure and the rictional stress depend on each other. In order to solve rictional problems, the equations must thereore be solved together. I we simply superimpose the results or the cases with and without riction, there may be signiicant errors. Acknowledgements: The project was supported by the ational atural Science Foundation o China (Grant o. 5117518). Open Access: This article is distributed under the terms o the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Reerences [1] Wen S Z, Huang P. Principles o Tribology. Singapore: Weley and Tsinghua University Press, 011. [] Bhushan B. Introduction to Tribology. ew York (USA): Wiley, 00. [3] Moore D F. Principles and Applications o Tribology. London (UK): Pergamon Press, 1975. [4] Quan Y X. Engineering Tribology. Hangzhou (China): Zhejiang University Press, 1994. [5] Huang P, Meng Y G, Xu H. Tribology Course. Beijing (China): Higher Education Press, 008. [6] Pinkus O, Sternlicht B. Theory o Hydrodynamic Lubrication. London: McGraw-Hill, 1961. [7] Крагельский И В (Wang Y L, et al., translate). Principles o Friction and Wear. Beijing (China): Mechanical Industry Press, 198. [8] Bowden F P, Tabor D. The Friction and Lubrication o Solid. Oxord (UK): Oxord University Press, 1954. [9] osonovsky M, Bhushan B. Multiscale riction mechanisms and hierarchical suraces in nano- and bio-tribology. Mater Sci Eng R Rep 58(3 5): 16 193 (007) [10] Kapsa P. Tribology at dierent scales. Adv Eng Mater 3(8): 531 537(001) [11] Xu Z M, Huang P. Study on the energy dissipation mechanism o atomic-scale riction with composite oscillator model. Wear 6(7 8): 97 977 (007) [1] Kim H J, Kim D E. ano-scale riction: A review. Int J Precis Eng Manu 10(): 141 151 (009) [13] Williams J A, Le H R. Tribology and MEMS. J Phys D: Appl Phys 39(1): R01 R14 (006) [14] Falvo M R, Superine R. Mechanics and riction at the nanometer scale. J anopart Res (3): 37 48 (000) [15] Fujisawa S, Ando Y, Enomoto Y. Microscale riction and surace orce. J Jpn Soc Tribologis 44(6): 409 413 (1999) [16] Wen S Z, Huang P, et al. Interace Science and Technology. Beijing (China): Tsinghua University Press, 011. [17] Martin J M, Matta C, Bouchet M I D B, Forest C, Mogne T L, Dubois T, Mazarin M. Mechanism o riction reduction o unsaturated atty acids as additives in diesel uels. Friction 1(3): 5 58(013) [18] Samadashvili, Reischl B, Hynninen T, Ala-issilä T, Foster A S. Atomistic simulations o riction at an ice-ice interace. Friction 1(3): 4 51(013)
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