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Extension of Friction Analysis in Rotating Machinery: A Roughness-dependent Model of Frictional Torque Occuring in Contacting Surfaces of Axis-symmetric Geometry
I. Introduction:
Friction is a universal phenomenon occured during the relative motion of bodies in contact. Friction usually causes energy dissipation which often result in the loss of mechanical efficiency thus changing dynamics of the system. [revise]
In most application, friction is modeled with a constant coefficient of friction, which is the quotient of tangential frictional force and applied normal force. This is also known as the famous Coulomb model. [cite] Particularly, in a machanical system which involves contact between surfaces of solid, the friction coefficient is usually pre-determined through various empirical methods, and is then used extensively for futher study [cite]. In rotating machinery, finding the formula for cumulative frictional torque is frequently of great interest since it is key to the analysis of power loss, efficiency, etc.
Traditionally, friction between two surfaces in relative motion is investigated by modeling the surfaces perfectly smooth with known geometry (which are frequently represented by continuous functions), allowing the rapid evaluation of applied normal forces in both magnitude and direction as functions of location (or alternatively called distribution). Thus, one can readily find the distribution of frictional force by implementing the Coulomb model and the cumulative frictional torque by simply integrating the distribution over the entire surface with boundary conditions. The cumulative frictional torque – applied load relationship in this smooth surfaces model is thus always found to be linear.
[review literature? What have been done?]
However, contacting surfaces are non-smooth in nature and will always presents some irregularities (often called asperities) [cite]. Furthermore, these irregularities are rather time-variant, inherently altering the surfaces’ geometry and accordingly the distribution of friction especially when the change is dramatic (for example during the running-in procedure). The varying distribution of friction can potentially lead to different values of the cumulative frictional torque, which cannot be observed in the smooth surfaces model. Thus, it is imperative to create a mathematical model built upon a non-smooth geometry in 3D, which allow us to quantitatively analyze the impact of surface’s non-smooth property on the cumulative frictional torque.
This paper will focus on studying the rotating pair of general axis-symmetric geometry (as seen in bearings, pads, breaks, etc.). The preliminary knowledge of surface’s geometry and contact mechanics will be presented in Section 2. The main part of this study which is the formulation of cumulative frictional torque expression as function of surface roughness, applied load, geometry, material properties, etc. will be presented in Section 3. Section 4 shows the application of this expression in some common geometries, which is followed by further analysis and experiments in Section 5.
This work extends the traditional method of frictional torque analysis, which has not been accounted for the presence of surface roughness. A detailed framework for the calculation of roughness-dependent frictional torque is created. The scope of this formulation is applied to any contacting surfaces with general axis-symmetric geometry. Insight to the frictional torque – applied load relationship is given using this extended model: as surface roughness increases, the dependence of frictional torque on applied load becomes more nonlinear, which is further validated through experiments.
 
II. Modeling non-smooth surface contact:
II.1. Surface structure and topography: [Menezes 6]
The surface structure may include roughness, waviness, lay and flaw [Menezes 6]:
Roughness: Roughness is produced by fluctuations of short wavelength characterized by asperities (local maxima) and valleys (local minima) of varying amplitude and spacing.
Waviness: Waviness is the surface irregularities of longer wavelengths. It may result from such factors as machine or workpiece deflections, vibration, chatter, heat treatment, or warping strains.
Lay: Lay is the principal direction of the predominant surface pattern.
Flaws: Flaws are unexpected and unintentional interruptions in the surface structure caused by dislocations and other material defects.
When a surface is stated as non-smooth, it can include all of those above features. In the scope of this study, only roughness is considered. While the waviness, lay and flaws are generally related to the manufacturing techniques, roughness is unavoidable in nature due to the particle-based structure of matter [Menexes].
Various roughness parameters have been introduce to evaluate some of the real surface geometrical characteristics. Surface roughness parameters are normally catagorized into three groups according to their functionality [Menezes]:
· Amplitude parameters
· Spacing parameters
· Hybrid parameters
(fig)
 
 
 
 
 
 
 
II.2. Rough surface modeling in contact problem:
Various models have been proposed to study the rough surfaces in contact, which are either deterministic or mathematical. If a model is considered deterministic, it would attempt to describe the entire surface geometry by realistic measurements of the interested surfaces. The disadvantages of deterministic models are that they are very computationally expensive and cannot be generalized, although they can yield more accurate results. On the other hand, mathematical models which implements different methodology such as statistical, fractal on multi-scale can overcome the drawbacks of the deterministic models, while compromising only a little accuracy in some particular applications [Menezes].
One pioneer work in modeling rough surface mathematically is by Greenwood and Williamson [124, [3]], who used statistics to describe the "random" nature of surface roughness. Their work is later used extensively in contact mechanics and is frequently refered as the GW model. Consider two surfaces that are in contact. One surface is perfectly smooth and is located at some distance  (from now refer as the clearance) from the reference line . The other suface is considered rough with the asperities’ height  is normally distributed around the reference line and is described by the Gaussian distribution (Skewness = 0 and Kurtosis = 3) (fig):
 
(1)
where  is the RMS roughness parameter of the surface.
Further assume that each asperity's deformation is independent of each other’s deformation and its peak is spherical with curvature (fig).
If the total number of asperities on this surface is , the number of asperities having height in the interval  comes into contact is  and thus the total number of asperities in contact is  = .
Furthermore, according to the Hertzian contact theory, when a elastic sphere is indented to depth  (from now refer as the indentation depth) in an elastic half-space:
The contact area is  and the required force is .
Thus the cumulative contact area is  and the cumulative required force is  with  is the equivalent modulus of elasticity and can be found using  ( are the moduli of elasticity and Poisson’s ratios of the two bodies).
If a thin layer of coating is present, a different equivalent modulus of elasticity should be considered following to Liu [Menezes112].
Very recently, the exact solutions for these forms of integral have been found by [Jackson & Green, 2011 - "On the modeling of elastic contact between rough surfaces"]:
If  and , then:
 
(2)
 
 
 
 
 
 
(3)
where:
,
 and  are the modified Bessel function of the first & second kind respectively
 is the complimentary error function
 is the gamma function
 
III. Formulation of expressions for cumulative frictional torque:
To approach the final expression for cumulative frictional torque on the entire surfaces, friction at a single asperity contact followed by friction on a differential area are investigated by applying the GW model. The GW model requires the number of asperities and the indentation depth at each asperity to be found. Furthermore, the magnitude, direction and location of application (in other word, this is a bound vector) of reaction force at a single asperity contact should also be expressed mathematically. What is known is the applied load, geometry and material properties of considered surfaces, and therefore every expression should be be writen in terms of these given parameters.
III.1. Differential geometry of surfaces:
i) Vector formalism of line in 3D space:
It is very convenient to express a general bound vector in 3D space using vector formalism. A bound vector is completely defined if its magnitude, direction and initial point are specified. In this problem, three quantities need to be expressed vectorially are the asperity’s direction, the reaction force and the friction force.
Consider a straight line L is defined by two parameters  (fig). An arbitrary point Q on the line has position vetor  is given by  where is an arbitrary scalar.
We are also interested in the point where an asperity comes into contact: a point  is the infersection of two line L1and L2has position vector  given by:
 
(4)
 
ii) Differential geometry of surface of revolution: [Grey 1997 wolfram]
To describe what happen at the contact of any asperity on the surface, differential geometry is used to describe many of the surface’s characteristics as function of surface’s parameters.
A general surface S in 3D space can be generated by two parameters  and . Any point on the surface has the position vector . In orthonormal coordinates the surface of revolution can always be expressed as: [Grey 1997]
 
(5)
Let  and :
The unit normal vector  of the surface is:
 
(6)
The area of a differential surface element is:
 
(7)
For a surface of revolution, it is natural to pick
If a surface is obtained by rotating the curve from  to  around the , from Eq. (5) any point on this surface has the position vector:
 
(8)
with  and . (Có hình vẽ)
From Eq. (6) and Eq. (7) , the corresponding unit normal vector  and the corresponding area of a differential surface element  is:
 
(9)
(10)
Furthermore, if , the (apparent) area of this surface given by Anton, 1999 is:
 
(11)
 
III.2. Formulation of general expression for the cumulative frictional torque:
Consider a surface of revolution (1) with its vertice  is at the origin of frame  and a surface of revolution (2) with its vertice  is at the origin of frame . We can always choose  such that the position vector of  with respect to  is the eccentricity vector . Because the surface is axis-symmetric and the asperities are randomly distributed, we can further assume that the eccentricity vector is in the same direction of the load , which is a known vector (fig).
(Có hình vẽ)
According to the GW model, let S1 is a rough surface and S2 is a perfectly smooth surface. The geometries of S1 and S2 are generated by rotating the curves  and  around respectively.
Consider a differential area at point  on the rough surface S1 defined by two parameters  and associated with normal vector  (resolved in ).  intersects the smooth surface S2 at point . is normal vector of the smooth surface S2 at point  (resolved in ). From Eq. (9) and Eq. (10):
 
(12)
 
(13)
 
(14)
Introduce  and with  is any point on the line . Physically, , ,  represent the height, the direction of a single asperity and the direction of reaction force respectively (fig). System of equations for the intersection at  can then be derived from Eq. (4) and Eq. (8):
 
(15)
Solving this system, we obtain  as well as , . Substitute , into Eq. (13), we could express  in terms of  and . The indentation depth of a single asperity can be found as (fig):
 
(16)
where .
In a differential area, the asperities can be assumed to be unidirectional. If the density of asperities is , the number of asperities in that differential area is . Since the asperities’ height is normally distributed described by the Gaussian distribution, the number of asperities on a differential area that height in the interval  is:
 
(17)
Furthermore, the position vector of a single point of contact and thus the moment arm respectively are (fig):
 
(18)
Assume the relative angular velocity is , the directional unit vector of the frictional force at an asperity contact is:
 
(19)
Finally, we attain the expressions for the number of asperities, real contact area, reaction force components and frictional torque in terms of a single asperity, a differential surface area and the entire contacting surfaces:
 
Single asperity
Differential surface area
Entire surface
Number of asperities
in contact
1
(20)
Real contact area
(21)
Reaction force
 
Reaction force
(x - component)
(22)
Reaction force
(y - component)
(23)
Reaction force
(z - component)
(24)
Eq. (20) shows the number  of asperities that are in contact. Eq. (21) can be solved using Eq. (3) that approximates the real contact area. Eq. (22), (23), (24) yield the components of the cumulative reaction force which according to Newton’s third law have to be equal to the components of the applied load. Finally, Eq. (25) gives the expression for the cumulative frictional torque that we are interested.
 
Frictional torque
 
Table. Expressions for the number of asperities, real contact area, reaction force components and frictional torque
 
 
   (25)
IV. Exact solutions for some common geometries:
IV.1. Spherical annulus with constant clearance:
Consider two spherical surfaces defined by the revolution of two functions: Có hình vẽ
 
(26)
 
(27)
with:
 
(28)
      
(29)
 
     
(30)
Dropping the subscript of . We can readily verify that and:
 
(31)
Thus from Eq. (16) and Eq. (18):
 
                                
(32)
 
(33)
 
(34)
Substitute Eq. (1), (26), (32), (34) into Eq. (25), we attain the cumulative frictional torque expression:
 
(35)
where:
 can be solved using Eq. (2).
To solve for
 
 
Even though the exact solution for Eq. (35) can be found, it is rather lengthy therefore will be presented later in the appendix.
 
IV.2. Eccentric cylindrical annulus:
Consider two cylindrical surfaces defined by the revolution of two functions:
 
(36)
 
 
(37)
with:
 
(38)
 
               
(39)
 
(40)
Eq. (15) then becomes:
Squaring  and  then take their sum:
After solving for  and dropping higher order terms of , we get:
 
(41)
Solving for  then substitute into Eq. (12) and Eq. (13), we get:
 
                      
(42)
 
(43)
Thus from Eq. (16), (18) and (19):
 
       
(44)
 
(45)
 
(46)
Substitute Eq. (1), (35), (44), (46) into Eq. (25), we attain the cumulative frictional torque expression:
 
(47)
 
V. Analysis and experiments:
 
We explore the dependence of the cumulative frictional force on the surface roughness parameter () and the applied load (). ------- compare to the conventional model
 
 
 
 
 
 
 
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