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Hello and welcome to 37 th lecture of video course on Tribology, topic of present lecture
is a Friction and Lubrication of Gears, this is the second lecture in this series. In previous
lecture also we have same topic, lubrication friction and lubrication of gears. We mentioned
in previous lecture, that there are two sources of friction, when we discussed about the gears.
And those sources will decide what will be the efficiency, that is why if we do that
kind of analysis, we will be able to figure out what will be the efficiency and that lubrication
part is going to decide what will be the durability of the gears, what will the survivability
of the gears, whether the gears are going to wear out or not. And if they are going
to wear out, what will the rate of wear? That depends on lubrication, kind of lubrication
mechanism we are using. And we discussed about the rolling and sliding,
we say that the pitch circle diameter particularly in spur gears, there will be rolling motion.
And in rolling, mostly there will be elastohydrodynamic lubrication mechanism, because the pressure
generated at the rolling contact substance is very high, substantially high, here we
say more than one gigapascal. And when the pressure is more than gigapascal, naturally
there will be thickening of grease or liquid lubricant whichever we are using, that technique
will convert semi solid to the almost solid and liquid to the semi solids.
In addition, there will be elastic deformation that is why we say that elastohydrodynamic
lubrication mechanism. In addition, there will be sliding, when we talk about the pitch
circle diameter there will be rolling, but below base circle diameter and above pitch
circle diameter there will be some sort of relative sliding, relative velocity of the
interface, that will be causing some sliding or friction of the surface.
So, we need to control that sliding using lubricants, then we discussed something about
the solid lubricants, say we can use solid lubricant something like nylon, we can use
a solid lubricant like polymers, we can make it gears directly from the polymers, then
we do not require extra lubrication. But depends whether we have application of
light duty, medium duty or heavier duty, medium duty load application and heavy duty of load
application, we will not be able to utilize polymers in those situations. So, polymer
materials are suitable only for the light duty applications, for heavy duty we need
to use some sort of liquid or semi solid lubricants. Solid lubricant coding can be done, but it
will wash out and we need to replenish one way or another way.
We discussed about major factor in a, as a tooth sorry toothing power loss, we say that
toothing power lose can be divided into sub categories; one is called ideal power loss,
that means there is no load applied on the gears, but still there is a speech variation
or speech change that will cause churning effect.
We can see this kind of the tooth profile and we assume this whole tooth profile is
submersed in liquid, what will happen? There will be a some sort of churning effect, some
sort of a splashing action, when the this wheel starts rotating and that is splashing
is going to consume power, that will not be without power. So, that is ideal power loss
and this ideal power loss will be higher when the speed is higher in other words churning
loss, loss will be higher with increase in a speed, it will heavily depend on the speed,
other one is a load power loss. We say that this kind of power loss will be depending
on how much load we are applying; more and more load more and more power loss.
To simulate this, what we use the relation something like this, the payloads equal to
F r into V g, this is the sliding velocity. This requires a sliding velocity with a rolling
velocity we are not considering there is a power loss, and this is the force which is
being applied over other contact surface or it is a friction force and that can be represented
as a cultural friction into normal force, normal force on the surface. So, P load which
is required power due to the power loss, due to the load will be given as the P load equal
to mu, the coefficient of friction, and normal load of the interface into sliding velocity.
We know sliding velocity will be 0 at the pitch circle diameter, so the pitch, this
power loss will be 0. Other thing is that, whether this mu is remains constant or not,
W N will remain constant or not V g will remain constant or not. In actual case, mu is highly
variable will change with the radius of curvature, it will be depending on the geometry, it will
be depending on the normal load and V g is a sliding velocity.
Now, sliding velocity itself will change continuously, it will be maximum at extreme of gear tooth,
will be 0 at the pitch circle. So, depending on what is end of phase or what is the contact
point, that sliding velocity will be decided or in another word, this power loss due to
load is highly variable, it is continuously changing with position to position, it will
not be a constant value. So, we need to take some sort of mean value.
Now, we taking about the mean value, naturally we need to find out the mean value of the
coefficient friction also, that mean value of coefficient friction often is represented
as number of variables that is a normal load. If the load is very high, there is a possibility
of converting from full film lubrication to the elastohydrodynamic lubrication to the
mix lubrication and to the boundary lubrication. And we know very well, co efficient of friction
will be different in all four regimes, or in other words, gear may experience also regimes,
that will be dependent on the speed as well as normal load which is been applied to the
gear interface. Now, when we taking the gear load, naturally the phase width will also
matter, because the stress which are generated, plastic deformation or elastic deformation
that is generated will depend on the phase width also.
In addition, there will be rolling velocity, we say that pitch, V pitch is the rolling
velocity, rolling velocity, pinion rolling velocity of gear or summation of this rolling
velocity, is going to affect the mu value, then comes the comes the viscosity eta. So,
this viscosity higher the viscosity, there is a more possibility of higher the value
of mu m. But this mu m will depend on the range, there is a possibility that we take
very thin oil almost negligible viscosity, but that is going to cause gear to operate
in boundary lubrication; in that case, mu m will be higher.
So, overall we need to see the package, we need to see the what is the factor, how it
is depending, this is there is no direct relation between mu and eta, but there will be indirect
relation depending on the load regime, then comes surface roughness of the gear and pinion
surfaces, that is why we are writing the R composite it is a surface roughness or composite
surface sharpness. Now, when we discuss about normal load in
previous lecture, we mentioned that there is a contact ratio, we discuss about the contact
ratio. Say, if the contact ratio is more than 1 that means some time of the rotation to
get teeth pair are coming into contact. And because of that, there will be change in the
load quiet possible during engagement, load slowly increases reaches to the steady value
and after the time of the disengagement again the, this load will start decreasing.
So, when gear tooth will going to experience this kind of load variation, we need to use
that kind of the load when we try to find out the, what will be the coefficient of friction
and another thing, there is a possibility of change in a co efficient of friction itself,
because when we are talking about the, one few degree rotation of the gear wheel, we
are able to see the coefficient friction itself is going to change.
It may vary little bit depend on surface roughness, depend on the other geometry is, depend on
material film, but it will reach to the 0 value at the pitch circle diameter, the mean
value that is what the 0 value. Before that, there is a higher value and after that there
will be higher value, but the pitch circle diameter this coefficient friction is going
to be 0, that is why we need to model it properly, it should come with this kind of characteristics.
May be initially, there is a some sort of modulation, some sort of variation, may be
statistical thing, but it will come to 0 at pitch circle diameter, when we are talking
about spur gear, helical gear, of course when we talk about the high powered gear, again
in those situations this velocity or this coefficient of friction will not be 0, there
will be sliding and that will cause some sort of, some value of coefficient of friction.
So, we discussed about the normal load in our previous slide. Now, we are going to discuss
about the rolling velocity V pitch. Now, V; this is the summation of the rolling velocity
of gear and pinion, and it has been observed with high rolling velocity, coefficient of
friction will decrease. With high velocity, co efficient of friction is going to decrease,
what is the reason for that? See, I am assuming these are the gear wheels, maybe I can assume
this is the pinion, this is the gear and they are rotating and what will happen, and there
will be some sort of tangential velocity generated at this interface.
And this having some sort of profile, see it will get some sort of liquid over here
and with a rotation, dragging action of the liquid will happen, and more and more dragging
action will happen when there is a more higher and higher velocity, or we say that this is
more like a pumping action, this oil, this liquid lubricant is going to get pumped from
this side to other side right and that is going to generate more and more lubrication
or we say that, if initially there was a boundary lubrication or lubrication. Because of this
pumping action, liquid will be at the interface and then it will be elastohydro lubrication
or full film lubrication or mixed lubrication entirely depending on the load, how much load
we are applying. This is the why we are showing this kind of
curve as the coefficient of friction as V e which is summation of rolling velocity of
gear 1, rolling velocity of gear 2 is increasing coefficient of friction is going to decrease.
Of course, their final limiting value, it will not be continuously increasing and often
this coefficient of friction is represented in terms of V e, we using this relation we
say that mu is proportional to V e power minus 0.45 that is the inverse relation, we say
that this is advantageous, this figure shows higher and higher V e will be better and better
results for us, better and better efficiency, but this is what we are talking about the
load. When we discussed about the load, there is
a load dependent friction, there is other friction also, what is known as a spin loses,
churning loses, that happens because of the spin velocity of the gear and they have this
kind of profile, naturally dragging action will be there, and higher dragging friction
will generate higher friction loses or higher spin loses. And many times, high speed gears,
spin loses are far more or far greater than load dependent friction loses, but low speed
larger side gear, this gear this load dependent coefficient friction is generally higher compared
to the spin loses.
Then, we discussed about, we discussing about composite or surface roughness profile and
we say this composite surface can be given as the R q 1 square plus R q 2 square and
square root of this whole, this summation. R q 1 is the surface roughness, may be say
pinion R q 2 is surface roughness of gear. We need to use this for calculating power
composite or we say R composite can be given as a square root of R q 1 square plus R q
2 square. Now, it has been observed that when we manufacture
the gears and start operating those gears, they have a different surface roughness compared
to what we get after certain duration and that will be use the word running in time,
this is shown over here, the milling operation, rubbing operation, grinding, lapping, honing.
So, these milling and hopping generally give manufacturing processes, grinding, lapping
and honing are super finishing operation to the remove the asperities from the surface,
but we are able to observe, here the surface roughness is the 2.3 to 4.6 micron. So, it
is very very high surface roughness and it cannot be used as it is, that means after
milling, we require some sort of super finish operation.
However, if we are using this milling operation and may be, say for lower speed operation
gears, we are using this. After running, may be say couple of hours or 100 of hours whichever
the running in time, for the gears what we get is the lesser surface roughness is almost
half of the surface roughness. So, what we use finally, when mathematical simulation
time, we need to use a running in surface roughness, we cannot use start surface roughness,
because that is not going to give reliable results and we cannot really operate also.
When the surface roughness is generally high, we cannot gears of the higher speed, we need
to operate at the lower speed and remove the oil whatever the on the surfaces and change
the oil and then come to this so that we can do this analysis.
So, surface roughness has importance, you say we need to take care surface roughness
after running in duration. And for detail, you can refer to this book W J Bartz, The
Lubrication of Gearing as a complete book on lubrication of the gearing and we say that
this topic.
Now, what we talk about the gear surface roughness is that, highly related to film thickness.
When we are trying to decide which kind of lubrication, lubricant regime is going to
work going to work on that case. So, this is an equation elastohydrodynamic equation,
h min is the minimum separation between the gear pair which generally happens in some
micron level and this is radius of curvature or the interface, it gives in a load parameter,
velocity parameter, material parameter and load parameter which we have earlier discussed
when we were discussing about the EHL elasto hydrodynamic lubrication mechanism. This depends
on the load, depends on the phase width, here is the slight change there when we were discussing
earlier, we used to, we were using the symbol F for the force, in case of the gears we are
using the symbol F for the phase width, or we say that depth of the gear, width of the
gear, E prime is a effective young’s modulus, R prime is effective radius. We assume that
their two surfaces and both are having some radius of curvature.
So, we need to find out effective radius of curvature, coming to the velocity parameter
we say that it is depending on the viscosity, it depending on the velocity, summation of
velocities, again it is also depending on E prime and R prime. Finally, this is the
material parameter, it depends on pressure viscosity coefficient and E prime effective
young’s modulus. So, we need to account these factors, what
is the sensitivity, what we are able to see, they have cylinder, they have been simulated
when they drive this kind of relation and of course, we can say that when we refer different
books, different literature, everywhere this relation is slightly different, may be somewhere
you will find this in a 0.073 or multiplication factor will be 2.65, this velocity will be
in a power 0.67. So, there will be delta variation depending
on the experiments performed and this is the profit equation, we are not solving partial
differential equation to give this kind of relation, this is more like profit equations
and wherever the results are matching, they give coefficients according to those experiments.
As there will be always slight variation form one experiment to other experiments, and what
I have observed, overall results are not going to be drastically changed, there will be delta
variation, but that is within permissible limit. So, once we know these factors, we
can find out what is the film thickness. Now, once we know the film thickness, we can
find out its specific film thickness which is going to decide with a gear is under lubrication,
mix lubrication or elastohydrodynamic lubrication. I am leaving the possibility of full film
lubrication, there is gears will not operate from full film lubrication, at most they can
be operated in elastohydrodynamic lubrication, but we have a this kind of a division, we
say that when the lesser than boundary, lesser than 1, we need to say that this is the boundary
lubrication, there will more and more asperities in contact and may be having some productive
layer lubrication layer of heard of nano meter to the micron may be say lesser than micron
1, 2, 3 is a mixed lubrication where the boundary lubrication is also existing under some external
full film lubrication is also extending. And this lambda greater than 3, we have seen
the full film lubrication, but it is a elastohydrodynamic lubrication plus full film lubrication or
we can say that elastohydrodynamic lubrication can be treated as a full film lubrication,
because there is no contact. Generally, on the rolling surface near pitch circle diameter,
this kind of a pitch surface this is possible when there is a high pressure and is deforming
the surface and is the lesser surface roughness. So, this regime depends on both first is h
min, there is the minimum film thickness minimum separation and depends on the surface roughness
profile and what we have seen in previous slides that, if manufacturing process is not
very good, what we are going to get high surface roughness and surface roughness may be the
3 micron or 2.3 micron or 3 microns. Even after running in time, 2.3 micron and
this is also 2.3 micron that means we need up keep the film thickness more than 5 micron
between the gears. And if the film thickness is more than 5 micron, there is a possibility
of lesser positive drive and that will also, again in those situations will be losing the
efficiency from the gear, we want to operate gears as close as possible, but we want to
avoid the wear also, it is more like a trade off.
We want efficiency, positive drive, we also want no wear and that is why we need to have
some sort of trade off. And it has been observed that as this lambda is of we say capital lambda
is increasing, when it is reaching to the 1, there is a sudden change in the relative
life. If I compare when the lambda is 1.5 or with the lambda is 1, we found there is
a substantial change or in another word boundary lubrication cases, generally gear is not going
to give us very good life, reliable life, and that is the reason why we use the lubricant
additives in gear oil. Whatever oil we use, we need to use some sort of E p additives,
anti wear additives on the, in those gears, if you are not able to use those gears additives,
naturally performance will be lesser; it will be towards the boundary lubrication.
And gear life will be much lesser or relative gear life will be much lesser, but if there
is jump from 1 to 1.5, we are able to see there is a jump, substantial jump, significant
jump in relative line. So, that is important for us to find out what is the surface roughness,
find out what is the film thickness, we see here film thickness itself is depending on
number of parameters, if we increase the viscosity, film thickness is going to increase.
If we decrease the load, film thickness is going to increase, but sensitivity is different
whether here sensitivity is roughly 0.7, while here the sensitivity is 0.13. So, there is
a significant difference of 5 times difference in power. So, we need to see whether what
is the more suitable, increase in the speed or increase in decrease in the load whichever
is more beneficial, we should go ahead with that, but again the trade off will come whenever
we are increasing the velocity. Naturally, there will be more churning losses,
there are more churning losses, there will be power loss, and we need to have some trade
off in that. And whatever we have discussed in this, these are the fixed values so that
in this case, we are using the viscosity, we are using the value of alpha, they are
constant temperature values and we know alpha may not be that high temperature sensitive,
but viscosity will remain high temperature sensitive or sensitivity of viscosity is very
high compared to alpha. As the temperature increases, this eta is going to change, and
that can be given by number of relations, can be predicted, we took a discussed about
two common literature, two common relations; one is the voltus relation other is vogel’s
relation, we know the vogel’s relation required three constants while voltus relation which
is given over here requires only two constants. Again, when we see the voltus equation from
a different book, sometime we get this constant as 0.6, 0.7 and 0.8, but we know very well
whenever we are discussing about the gear oil, this viscosity, kinematic viscosity represented
as centistokes, there will be in numbers, may be say 30, 40, 50, 60, 70, 80, 90. So,
when we are talking about the 80 plus 0.8 is not going to make much difference when
you are taking the longer longer scale. Of course, this is the longer scale, means
we know the viscosity of the two different temperatures 40 and 100 degree centigrade,
we can find out the value of a and value b and once we find this, this kind of relation
can be used iteratively to find out what will be operating viscosity.
Question comes, how to iterate this, we do not know the temperature, we have not discussed
anything about the temperature, we are saying mu depends on regime, regime depends on the
film thickness, and surface roughness. Now, film thickness depends on viscosity, viscosity
is depending on the temperature. So, number of equations need to be solved together is
a highly complex, I mean good computer program will take may be say 5 to 6 hours minimum
to give one result.
So, that is why many times what we say that we will use some sort of physical modeling
or understanding based modeling, we know what is happening, we try to modeling on that.
So, these are the couple of diagrams over here, we say that when we try to do modeling,
we know there will be serial velocity here. The coefficient of friction turn out to be
0 over here, there is almost 0 sliding, but here there is a more sliding, we say gear
tooth can be divided at the pitch level, above pitch, below pitch, this flank and face, generally
the flank will be having higher wear rate compared to the faces.
That is why we say that, this portion is more subjected to fitting, is more subjected to
the value compared to this portion, and we need to give very good thickness or thick
over here. Similarly, when we talk about the sliding, here the sliding is much wider compared
to over here and that is going to give the friction surface and this is what we say is
more critical to analyze, and here importance comes from the role of the lubricant or we
say lubricant gets important over here. If we use lubricant, first thing even in the
rolling action, it is going to separate, it is going to generate elastohydrodynamic lubrication
mechanism, it is going to separate two surfaces. So, lubricant is important over here. In addition,
when there is a sliding, naturally lubrication is important, sliding is going to join in
a higher friction and to reduce that friction, and we require lubricant.
But problem is that, liquid lubricant itself gets shared. So, it will give some sort of
coefficient friction, some friction, but when we compare solid to solid, sliding friction
with the liquid, solid sliding friction, you find substantial change, that is why we prefer
liquid sharing compared to the solid sharing right. And here, it is also mentioned whatever
you do as we are not going ahead with full film lubrication and even elastohydrodynamic
lubrication happens only near to this surface, this circle, that is why there will be some
wear and that wear can be given be sliding velocity, can be given by load, normal load,
given by the material and hardness, material constant and hardness right.
So, it is heavily dependent and now we choose a bad material having lesser stiffness, naturally
the wear out will be more. If we apply more load, wear out will be more; if I apply more
sliding, wear out will be more. So, many times we say that we can compare this as stuffed
teeth, standard teeth and high contact ratio gears.
We say that when the, this profile or we say that this profile is extending beyond certain
limit, that is going to give the high contact ratio, but along with the high contact ratio,
it is going to every high sliding, high contact ratio is required from force point of view,
higher the ratio lesser will be the force, obviously that lesser will be stresses.
But we have the contact ratio; there will be more sliding; more sliding, more friction.
So, again when we do the friction analysis, we find there will be change and if there
is a wear of the gear, even though we keep a high contact ratio initially it is going
to decrease with time. It is not going to be constant, this contact ratio depends on
the time duration, there is a more wear surfaces are getting damaged and then contact ratio
will be decreased. So, in other words we say that gear will never
have infinite life; it will have only finite life depends with 20000 hours, of 30000 hours,
40000 hours. If you require more and more life, we need to require more and more wear
volume even though whatever you do the good design initially, but when talk about the
tribology, we need to reduce this wear rate, here wear rate can be reduced if you are able
to reduce the sliding velocity, we are able to increase the hardness, we are able to reduce
the load. So, one way or another way we need to do or reduce this k 1.
We know very well, k 1 is heavily dependent on the liquid lubricant, more and more lubrication
this k 1 should be, but larger and larger k 1 will larger and larger liquid lubricant
is going to produce more and more churning losses. So, again the trade off is required,
higher friction may lead higher temperature, that may lead more volume of the wear.
So, this is combination and that why we required some sort of procedure, we say determine effective
value of velocity, that is pitch circle diameter, you find out, you find out omega and then
you find out the value over there, then you find if you are able to calculate load and
normal load, then effective modulus of elasticity, and effective curvature, effective radius.
So, that is the first step, when you start we need to generally have a data available
to us, velocity can be determined based on the module number of teeth and that is which
is going to give the diameter and omega and then load can be figured out, can be calculated
if we know the power transmission or we know the torque applied on the gear.
That is in going to give results of the load, then modulus elasticity or effective modulus
elasticity, that require two material property, we say poison ratio and young’s modulus
and that is going to give the effective modulus of elasticity, then comes curvature. Here,
we are taking the curvature, this curvature is different, this is the curvature, we are
not talking about the pitch circle radius, we are talking about the convex shape of the
gear and that curvature is important. We need to find out either using pitch circle diameter
and pressure angle, you know the pitch circle, and the curvature radius can be calculated
using pitch circle radius and sine of the phi or sine of the pressure angle.
The multiple the product is going to give us curvature, then next one is the, to determine
and estimate effective temperature, we are not introduced many temperature relation,
we need to balance or we say go for the thermal balance, thermal equilibrium to find out what
will be the effective temperature, then once we know the temperature, we should determine
the viscosity. As we know, viscosity is going to get affected
with the temperature, higher the temperature higher, lesser will be the viscosity and if
possible, calculate pressure viscosity co efficient as a function of temperature, this
also may reduce, then calculate the film thickness. Once we know the film thickness, compare it
with a roughness and decide what will be the regime; once we know the regime, we can find
will be the coefficient friction. So, coefficient of friction may be on a, if
it is a mixed lubrication it may turn out to be lesser than 0.1, if it is a module lubrication
turn out to be 0.1 to 0.3. So, depends on the regime and number of parameters, we can
decide what will be the co efficient of friction. Again, we have given a bigger expression a
big dependence, but those are indirectly depending on the coefficient of friction and indirectly
affecting the coefficient of friction, what we say that we require so many calculations,
that is the one way.
But we can go ahead with other way, we can find out the effective temperature or maximum
temperature, or the flash temperature of the gear interface using this relation, say what
is this relation, we say his relation is been derived from rate of heat generation.
How to give the heat rate of heat generation, there is a mu which we are trying to figure
out, this is sigma d that is the stresses which are causing the deformation. And this
is the relative velocity at any time, what will be the velocity of the pinion velocity
of the gear and then we are going to take the difference, naturally this velocity will
be 0, this difference in velocity will be 0 at the pitch circle.
That is why we are not saying that, this T f will come at any time at the pitch circle;
it will be away from the pitch circle. It will be away from the pitch circle, the coefficient
of friction is also will be different, the pitch circle it will be 0. Now, coming to
the sigma d which is compressive stresses generated in the gear profile, we can see
these are the convex shape and we can find out the radius of curvature of this surface,
this surface. Once we know that this is the radius of curvature, we can find out what
will be the effective radius of curvature that is R I.
Similarly, if we know the material of this surface, material of gear and pinion, then
we can find out young’s modulus which requires the poison ratio of the gear material 1 and
gear material 2. Once we have young’s modulus, effective young’s modulus, we have effective
radius of curvature, we can go ahead with finding what will be the sigma d, because
we know sigma d is a strong function of material properties and radius of curvature, which
have been discussed our the basic model when we discussed about the elastic deformation.
Now, this is sigma d and we say that will be parabolic distribution, pressure will be
maximum in the centre and lesser away, this continuous variation in velocity will happen
every point will have a maximum pressure and may be say after 0.1 degree, 0.2 degree, it
will have a lesser pressure. What we are, once we know, we calculate sigma
d and substitute in this, what we are going to get this equation in terms of normal load
and phase width; larger the phase width, lesser will be the maximum temperature of less temperature.
And in addition, what has been mentioned over here is, V 1 beta 1 and beta 2.
That beta 1 and beta can be defined as a co efficient of thermal contact, coefficient
of thermal contact for the surface 1, coefficient for thermal contact for surface 2, that is
beta 1 and beta 2 and what is this thermal constant, thermal contact co efficient? We
say that it depends on the conductivity, if I know the thermal conductivity material,
that is going to give can be represented as the lambda, is a small lambda and we know
it need to have some sort of dissipation and the specific heat which can be represented
as specific heat into density, we see as a specific heat and row as a density.
This is going to give us a beta value; larger the beta value, lesser will be the temperature
or we say that if material conductivity is high, it can dissipate the heat faster and
faster. Then naturally, there will not be accumulation of the temperature and there
is a no accumulation of temperature, naturally thermal equilibrium will be faster and we
will get a lesser temperature, overall lesser temperature. So, this maximum temperature
is depending on velocity or relative velocity. So, larger relative velocity, larger will
be temperature and then it is also depending on sigma d which is also given in the function
of load phase width, young’s modulus and radius of curvature.
And this w, small w is this patch contact length, which in our earlier relation, in
earlier lectures, what we represented this as a b or 2 b 2 into b, this is contact patch
length, was represented if you refer earlier lecture we will get a similar expression what
we are doing here, that is elastic effective radius of curvature, effective young’s modulus,
substitute we are going to get same relation. So, after substituting all, what we are getting,
this is some multiplication factor into coefficient of friction, if we are able to find the boundary
lubrication, mix lubrication and based on that lubrication, we can find out what will
be the temperature. Naturally, if the co efficient of friction
is 0.1 and if I compare with 0.05, then co efficient this temperature will be reduced
to half, reduction of temperature into half is significant. Usually, we should point out,
there should be more and more additives to at least maintain boundary lubrication and
it should be some carrier fluid which can carry these boundary additives and try to
keep some separation, that will give lesser coefficient friction.
Then comes the relative velocity, and this is given in the expression beta 1 into V 1
beta 2 into V 2. Now, think over extreme cases, naturally when we are talking about the relative
velocity, they will have negative velocity as well as positive velocity, what we are
going to talk about extreme cases, assume that V 2 is not there, V 2 is 0, when V 2
is 0 what we are going to get maximum value of this, and V 1 will turn out to be overall,
if I compare V from 0 to some positive value, this will give me maximum value of V 1 minus
V 2 will not be V 1, if V 2 is positive. That is the situation, this is V and here
also this will turn out to be 0, this will be square root of beta 1 into V and this overall
expression can be modified, if we want to find out the extreme cases in trip conditions.
That is going to turn out to be something like this, yeah in this case, V 1 divided
by beta 1 beta 2 has been replaced and we know V 1 can be represented as omega into
radius, that radius at any contact point, we are not talking about the pitch circle
radius, contact point, that can be given in terms of number of teeth module. That can
be given in those terms, because we know for the different gears are generally standardized,
when the gears are standardized, naturally they will have some module, and some gear
number of teeth and based on those, if this number r can be figured out can be calculated.
So, overall expression will turn out to be this omega, rotational speed divide by beta,
load divided by phase width, do not get confused many times we get F as the force, but here
we are giving F as the phase width, it is in m, it is not in Newton while W is in Newton.
Now, here m is m is a module, Z is the number of teeth and we are talking about this E prime
as effective young’s modules. So, if you want to keep low value of teeth,
what is required, we require a lesser speed, larger beta, larger phase width, lesser load,
lesser module, that is very good thing, lesser module; in this case, lesser number of teeth
and lesser effective young’s modulus while we know that increase in number of teeth is
going to reduce the load. So, they are having some sort of dependence,
when this dependence comes whatever the final comes, we need to take that as, that into
account. Now, here f z f z is depending on the number of teeth, that is the couple of
numbers are shown over in this table, we say Z is 17, this value is 0.1, 0.813. So, here
the number of teeth, increasing number of teeth is going to decrease the f z value,
that is why I say and these proportionally too, here we are increasing the number of
teeth, that is going to increase the temperature. Increasing Z is going to increase the T f
value, but increase in Z is going to decrease f z. So, this is more beneficial compared
to this, it is only having the power of 4 or 0.25 while this in case power is 1. So,
we need to compare it and wherever the benefit comes, we need to take a decision based on
that, what will be the number of teeth can be decide so, based on the temperature calculation.
We can find out the beta with thermal properties of the materials, we can find out omega form
the rotational speed, whatever is been given and if we can find out the temperature individually.
Temperature individually for the gear and individually for the pinion generally it will
be more for the pinion in this case should go ahead with the pinion surface in this case.
And that this coefficient friction depends heavily on surface roughness as well as the
film thickness, film thickness itself will depend on number of materials wears there.
So, we need to, because of clear design we talk about the gear design, it will not be
simple like this, it will be based on combinations, number of number of teeth, young’s modulus,
effective curvature load is P parameter. So, there is no one way rule that we need to have
only these parameters. It is always a trade off, it is always a combination,
it is overall package gives a lesser coefficient friction and higher coefficient friction,
and we know the temperature is high, viscosity is going to decrease. If viscosity decreases,
it is going to decrease the load carrying capacity. So, it is basically iterative scheme,
it is iterative scheme that is why, we try to utilize this kind of analytical expression
of expressions, if we are requiring detail calculation, it will take many iterations.
So, if we are designing initial level gears, we must always recommend this kind of equation
find out the T f find out the viscosity, again find out the what will be the load carrying
capacity, what will be the h minimum, because h minimum itself will going to depend on the
viscosity. And when that is depending on viscosity, h
minimum h minimum is going to affect mu and mu is going to affect the T f and mu is going
affect the T. So, that is always a perfect combination or we say we are going to get
some temperature which is heavily dependent on itself, it is say, and temperature is depending
on itself in another way. So, this is what we say about pressure dependent
coefficient of friction. We can calculate churning effect which is generally about the
high, if viscosity is high churning effect will be higher side and that it will be required
whenever there is speed of operation is very very high. But generally, we say the gears
are recommended, generally gears are recommended as speed reduction device, they are reduced,
used to reduce the speed and increase the torque.
When we increase the torque, so in those particular applications, I feel the load dependent, power
loss is more dominant compared to speed dependent module, speed defined on churning effect may
turn out to be lesser than 50 percent when we talk about the real application of gears.
Generally, we do not use the gears for high speed application, but if it is required,
then we should calculate the churning effects, otherwise using only the pressure load dependent
friction losses will be sufficient for us.
So, we will see how to calculate, we will just take one example to find out what is,
how to utilize this relation. See, there we know the co efficient of friction and at different
regimes, if it is coefficient friction in dry case and I mentioned about one of the
examples, that polymers can be used as gear material. When we use polymer as the gear
material, we should check, what is the validity of those gear material, that is why we are
taking possibility of scuffing of high temperature, mean the temperature is very high there is
a possibility of plastic deformation, plastic flow and two materials will get welded and
that is known as scuffing. This scuffing is one of the failures, we generally
do testing of this scuffing on various gear machines, but if there is a proper design,
proper design of the gear, proper design of the lubrication mechanism, there will be very
low probability of scuffing, but it is possible when we use some sort of polymer gear material,
we need to find out is there any possibility of scuffing.
So, example says, compare scuffing possibilities on nylon nylon, versus nylon steel gear pair,
here again we are using two nylon gear, but here we are using one, a steel gear and one
nylon gear. We are not talking about the steel, we are talking about the nylon steel and when
we compare the co efficient of friction, what we find, nylon and nylon co efficient of friction
is slightly more than nylon on the steel, slightly more. So, we can start with a dry
lubrication, we say there is no liquid lubrication and mu can be taken as it is, it is a dry
lubrication mechanism.
So, we can implement that and try to find out, may be we require some data for calculation.
So, for time being, we have taken the pinion gear teeth as a 17, module as the 3 mm, that
is going to give as a 51 mm pitch circle diameter. And if you want the radius, we can divide
by 2, that will be pitch circle radius, phase width in this case, what I have written b
is of a 30 mm, and omega is a 150 radiant to power second, while power which is been
transmitted from nylon gears is slightly lower side is 850 watts. So, based on the power,
we do can find out power and omega and pitch circle radius, we can find out what will be
the W and of course, I am assuming there the pressure angle will be 20 degree, without
pressure angle will be very difficult for us to get the results, then this is phase
width and this is by mistake it is written as b here, but it is b f.
So, this b is 30 mm or b is f and f is 30 mm, the module has been defined number of
teeth are defined and we can find out young’s modulus, effective young’s modulus based
on the material pair, we know the nylon nylon material pair, we know the other pair is nylon
steel and young’s modulus on the the materials are known to us, poison ratios are known to
us, we can substitute those values, we can find out and this table is going to give showing
the results. That is, when we talk about the nylon, nylon
effective young’s modulus is roughly 2000 mega Pascal or we say 2 gigapascal, while
when we talk about nylon versus steel, we are forgetting its roughly 4 gigapascal, otherwise
young’s modulus for this steel is very very high, it is almost 200 plus gigapascal. But
talking about the pair, naturally lower value will be dominating, that is going to decide
what will be the final value and that is showing the 4 gigapascal.
So, once we know E prime, we know Z here, we know m here, we know F, we can find out
W, we this omega has been given to us, and coefficient of friction from previous table
we can figure out as 0.2 nearby 0.2, even though we know that nylon is steel will give
slightly lesser coefficient of friction compared to nylon versus nylon.
But we can take same value; almost same value is 0.2, 0.2 to find out what is the relation
or what the overall temperature is, is there any possibility of scuffing, is there a possibility
of high temperature, high plus temperature. So, once we use this parameter, of course
there is a beta parameter which can be figured out from the material combination and here
we are able to see that. Beta for nylon versus nylon is roughly 0.417
10 is to 6, while in case of nylon versus steel, because of thermal conductivity. Because
of high thermal conductivity of steel, this is substantially high value, compared to 0.147
is at 180 almost 360, almost of the 360 times, very high value, high compressor and that
is in the denominator. So, higher value of beta will be preferred
to keep temperature lower side, and that is what is going to happen, you can see here
for nylon versus nylon, the temperature is the 161, very high temperatures for the nylon
to bear it, without. Of course, in number of within number of fillers, it can be increase
the number of fillers can be increased, but still we cannot say there will be elastic
deformation, there will be perfect plastic deformation, they are when temperatures are
this high, naturally contact will be subjected to slightly plastic deformation, it will not
be only the elastic deformation. While coming to the steel versus nylon, this
temperature is only 19.5 degree, very low value compared to 161, it is not only the
one eighth of that or 12.5 percent of that, but with increase in temperature, sensitivity
changes drastically. So, this combination will be preferred or we say that whenever
we choose the gears, may be one polymer gear, but other solid material gear or which has
a very good hardness, which can have, which has also the high thermal conductivity. If
the gear has hardness and thermal conductivity, surely we can use this kind of pair, one solid
material pair, one one solid material gear of pinion and other as a steel or some material
which has high thermal conductivity, that will be preferable compared to this kind of
thing. But we know very well, for the number of application
in home appliance, this nylon nylon gears are used, given the reason being their duty
is lower side, they do not transmit power continuously, and it has generally recommended
we talk about the mixture, we operate the mixture, we say we do not operate at 10 minutes,
because we continuously operate for 10 minutes, the temperature will continuously increase.
And we may find that, gear versus gear is going to get engaged or they will be folded,
it cannot be separated, it cannot be rotated after that. And that is the reason we want
to keep operation for some time, not beyond certain level, because they will be thermal
instability, there will be more flow and there will be more flow cold welding, and this is
a important aspect whenever we choose a gear material, we need to think about good thermal
conductivity, because we know the contact point there will be high temperature.
There is a possibility of high temperature and there is a metal to metal contact and
in this whole example, we took coefficient of friction is equal to 0.2, but if we provide
some sort of lesser coefficient of friction, that is going to affect the results. If I
decrease this coefficient of friction, may be say ten times, naturally this temperature
is going to reduce then 10 times. But in that, in those situations, there is nylon selecting
nylon will not be any use, it can be removed or we can choose some alternative material.
So, we will continue our gear topic in our next lecture.
Next lecture, we will be considering what kind of failure analysis we can do, or what
are the possibilities of gear failure, we have explored possibility of gear failure
in this case, particularly scuffing failure which happens at high temperature of the lubricant
or very high co efficient, because of the high coefficient of friction and lesser thermal
conductivity, but there are other possibilities, or possibilities of failure of gears, we will
be exploring in our next lecture, thank you.