Table of contents
Nonmenclature
Number the formulas
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):
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:
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
L1![]()
and
L2![]()
has position vector
![]()
given by:
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]
Let
![]()
and
![]()
:
The unit normal
vector
![]()
of the surface is:
The area of a
differential surface element is:
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:
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:
Furthermore, if
![]()
, the (apparent) area of this surface given by Anton, 1999
is:
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):
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):
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):
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:
Furthermore,
the position vector of a single point of contact and thus the moment arm
respectively are (fig):
Assume
the relative angular velocity is
![]()
, the directional unit vector of the frictional force at an
asperity contact is:
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:
IV. Exact solutions
for some common geometries:
IV.1. Spherical
annulus with constant clearance:
Thus from Eq. (16)
and Eq. (18):
Substitute Eq.
(1), (26), (32), (34) into Eq. (25), we attain the cumulative frictional torque
expression:
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:
Eq. (15) then
becomes:
Thus from Eq. (16),
(18) and (19):
Substitute Eq.
(1), (35), (44), (46) into Eq. (25), we attain the cumulative frictional torque
expression:
V. Analysis and
experiments: