In order to understand the true impact the automobile has had on our society,
we would have to go back in time over one hundred years. A time without the
simplicity of hopping into a vehicle to take us anywhere we want to go is almost
unfathomable to many Americans. But for the early automotive engineers, the
tremendous advancements in automotive technology would be even more
surprising.

In the last 50 years, cars have learned to think, adjust, and even protect.
But this is just the tip of the iceberg. High performance is now the catch
phrase. The vast majority of people want a vehicle that will get them from point
A to point B as easily as possible, but also put a little smile on their faces.
Often times, the smile is created by a quick punch of the accelerator and
accompanied by a feeling of immense power and control. The auto manufacturers
are well aware of this, and to achieve it, they design faster, lighter, and more
efficient engines to do the job. But exactly what happens inside an engine, and
what are the risks involved in designing the strongest engine on the block?

In this project, one component of an engine in particular, the connecting
rod, will be analyzed. Being one of the most integral parts in an engine’s
design, the connecting rod must be able to withstand tremendous loads and
transmit a great deal of power. It is no surprise that a failure in a connecting
rod can be one of the most costly and damaging failures in an engine. But simply
saying that isn’t enough to fully understand the dynamics of the situation.

Throughout the course of this project, an idealized model of a connecting
rod, piston, and flywheel will be modeled and analyzed. It will become apparent
exactly why these parts are so important to the operation of an automobile, and
furthermore how prone to failure they can be. However, before too much more is
said on the engineering details, a little background information is
necessary.

Now that we are all on the same page, the assumptions for this project can be
discussed. First of all, it is necessary to point out that the actual dynamics
of such a system are tremendous, and to model all of them in one project would
be quite a task. So, to simplify, this project will neglect momentum and
gravity. Only one connecting rod-piston assembly will be considered. The
crankshaft, while in actuality having a very functional mass distribution, will
be considered simply a circle (or if it is easier to visualize, a flywheel). In
effect, many of the same calculations could be performed on a more sophisticated
system, but this will suffice for the time being.

From an understanding of statics, we can represent the connecting rod of
length "l" by a two-force member (this requires a few more assumptions, but for
purposes of this project, it is acceptable). Given this, we can split this
system into two free-body diagrams:

From these free-body diagrams, we can apply Newton’s Second Law (F=ma) to
write some equations. In particular, we are interested in summing forces in the
"x" direction (horizontal), and summing the moments about the center of the
flywheel. Doing so, we acquire these equations:

S M_{o} = -F_{AB} cos (F ) * rsin (Q ) – F_{AB} sin
(F ) * rcos(Q ) = I *
d^{2Q }/dt^{2} (CCW positive)

S F_{x} = -F_{AB} cos (F ) – P = m * d^{2}x/dt^{2} (® positive)

We can simplify the moment equation, employing the use of the double-angle
trigonometric formula:

sin (F + Q ) = cos (F ) * sin (Q ) + sin (F ) * cos (Q )

Therefore,

-F_{AB} * r sin (F + Q ) = I * d^{2Q
}/dt^{2}

Now, if we solve the force equation for –F_{AB},

-F_{AB} = (m * d^{2}x/dt^{2} + P)/(cos (F ))

We can substitute this equation into our moment equation, giving us:

(m * d^{2}x/dt^{2} + P)/cos (F ) * r
sin (F + Q ) = I *
d^{2 }/dt^{2}

This will be our main equation of rotation.

At this point, we are working our way towards acquiring a representation of
Q , in order to eventually find F_{AB}. But
looking at these equations, we can see that there are many different variables
to work with, including a few derivatives. In order to help simplify them a
little more, it is important to notice a few relations. For instance, we can
apply the law of sines to this triangle, found between the flywheel and
piston:

sin (Q )/l = sin (F )/r ^{2}

This takes care of the two angles. Next, we must find an equation for x, the
distance from the center of the flywheel to the bottom of the piston. This can
be found using trigonometry:

x = l cos (F ) + r cos (Q
)

Unfortunately, we are not actually dealing with x in this problem, but rather
d^{2}x/dt^{2}. Therefore, we will have to take two derivatives
of x:

dx/dt = -l sin (F ) * dF
/dt – r sin (Q ) * dQ /dt

d^{2}x/dt^{2} = -l cos (F )*(dF /dt)^{2} – l sin (F
)*(d^{2F }/dt^{2}) – r cos (Q )*(dQ /dt)^{2} –

r sin (Q )* d^{2Q
}/dt^{2}

With this value for d^{2}x/dt^{2}, we can substitute back
into our main equation. However, once again we have introduced a few more items
into this scenario, in particular the first and second derivatives or Q and F . Given these terms, we will
once again have to find equations that relate them to things we already know or
can find.

Because this is a long process, I will explain what is happening beforehand
and then simply show the equations. We have our equation relating Q and F that was derived from the
law of sines. From this, we can take a few more derivatives to find equations
for dF /dt and d^{2F
}/dt^{2}. It is not necessary to find the relation of the
derivatives of Q because they will be shown in the
final integrations.

F = sin^{-1}((r sin (Q ))/l)

dF /dt = r sin (Q ) *
dQ /dt

l cos (F )

d^{2F }/dt^{2} = -r cos (Q ) * (dQ /dt)^{2} +
r sin (Q ) * d^{2Q
}/dt^{2} + sin (F ) * (dF /dt)^{2}

l cos (F ) l cos (F ) cos
(F )

**PRESSURE IN A FOUR-STROKE ENGINE**

Up to this point, the variable P has gone unmentioned. The pressure in the
cylinder (P) is not an easy thing to model for a situation like this, yet it is
one of the most important factors in the final analysis. To be able to explain
how P fluctuates, it is once again necessary to give a little background on a
four-stroke engine.

A four-stroke engine is the most common type used in automobiles. The four
strokes are intake, compression, power, and exhaust. Each stroke requires
approximately 180 degrees of crankshaft (or flywheel) rotation, so the complete
cycle would take 720 degrees. Each stroke plays a very important role in the
combustion process, and each has a different pressure surrounding it.

In the intake cycle, as the picture shows, the piston is
moving downward while one of the valves is open. This creates a vacuum, and an
air-fuel mixture is sucked into the chamber. This would be cause for very little
pressure on the piston, so P is small.

Moving on to compression, we can see that both valves are
closed, and the piston is moving upward. This creates a much larger amount of
pressure on the piston, so we would have a different representation of P in our
equation for this stroke.

The next stroke is the big one: power. This is where the
compressed air-fuel mixture is ignited with a spark, causing a tremendous jump
in pressure as the fuel burns. The pressure seems to "spike", so the most cause
for concern occurs here. (This is also the area in which the dangers of engine
knock or pre-detonation can occur, causing an even larger spike.)

Finally, we have the exhaust stroke. In this stroke, the
exhaust valve is open, once again creating a chamber of low pressure. So, as the
piston moves back upwards, it forces all the air out of the chamber. The
pressure in this region is therefore considered very low.

So, given the understanding of how a four-stroke engine works, we must now
model the variable pressure for all 720 degrees (or 12.57 radians). Creating a
piecewise-defined function does this. However, we still need to find some basic
values for the pressure, and for the purposes of this project, a particular
graphical representation was chosen:

In order to make this graph work, we assume all points are linearly
connected. In other words, three pressures were selected (5, 10, and 30 atm),
and it was assumed that the pressure increased linearly between them. With this
assumption, the piecewise-defined function became (angles in radians):

P = 10 +200*Q 0 < Q < .1

30 .1 £ Q < .35

30 + (20-57.14*Q ) .35 £
Q < .7

10 + (5-7.14*Q ) .7 £ Q < 1.4

5 1.4 £ Q < 11.87

5 + 7.14*Q 11.87 £ Q <
12.57

**INTEGRATION AND DATA ANALYSIS**

Now that we have everything represented in one way or another, it becomes
necessary to focus our attention on finding Q . Because
such a complicated equation cannot be solved analytically, a numerical method
needs to be used. In this particular case, given its complexity, the Euler
method of integration was chosen.

In order to perform all of the calculations, a program was
written in Fortran. Essentially, it asked for the mass, radius, and length from
the user, as well as the starting values of Q , dQ /dt, and d^{2Q
}/dt^{2} and the time step value, while producing the values of
Q and the actual force in the connecting rod,
F_{AB}. In order to analyze the data over a period of time, the Q and F_{AB} values were sent to a file, which was
read by Microsoft Excel and graphed over time. The two graphs, Q vs. time and F_{AB} vs. time, are displayed below.

Note: For purposes of this project, the following substitutions were made:
the length was set equal to 6.7 in, radius of the flywheel was 4.33 in, the
radius of the piston was 2.31 in, and the mass of the piston was 3 lbs.

**DATA IMPLICATIONS**

It appears from looking at the graphs that a mistake must have been made. It
would take far less time for a flywheel to rotate than the 10-15 seconds that
could be inferred from the graph of theta. In addition to that, we would also
assume theta to be constantly increasing as opposed to the rapid fluctuations it
experiences here. Unfortunately, it is very difficult to say what is causing
this error. There is always the chance of mathematical errors occurring, but it
could also have been due to the fact that the system started out from rest. In
actuality, a starter would give the flywheel an initial turn or two to get
things moving, letting combustion eventually take over. However, that becomes
very difficult to model in this case.

Generally, we would assume that because our graph of theta is incorrect, the
graph of force is probably as incorrect. However, looking at it, we see that
there are certain areas where the force is much greater. This is definitely
expected, but probably not to the degree that it shows up. But, for the sake of
discussion, let us just assume that the high value of around 550,000 lbs that
shows up on our graph is actually correct. Making that assumption, we can do
some calculations and relate this data to the Mechanics of Materials.

First of all, to find the stress on the connecting rod, we use the
formula:

s = P/A (s = stress, P =
force acting on rod, A = cross-sectional area)

If we assume a connecting rod thickness of approximately .5 in and a width of
1.25 in, we would calculate a cross-sectional area of .625 in^{2}.
Therefore, plugging in our values of P=550,000 lbs and A=.625 in^{2}, we
would get a stress of 880,000 pounds per in^{2}.

Now what does this value mean to us? Is this a lot of stress to place on a
material? Well, let’s assume that the two most likely materials to be used in a
connecting rods are steel and aluminum. This is actually a pretty good
assumption, since the vast majority of automobiles contain steel or aluminum
rods.

**STRESS-STRAIN DIAGRAMS**

To understand the strength of each material in a situation like this, we need
to understand a stress-strain diagram (pictured below). Each material behaves in
a similar manner when placed under a load. There is a period of elastic
deformation, in which the material is stretched, but it returns to its original
size when unloaded. The point at which it fails to return to the original
specifications is called the yield stress. Now, in an automobile, we would
probably have to assume that this yield stress would be passed at some point, so
most connecting rods come out of engines a different size than when they were
installed.

After the yield stress, another stress point can be reached called the
ultimate stress point. At this point, a material has essentially reached the
point of no return. Failure is imminent, and even a decreased amount of stress
can cause fracture. So, naturally, this is what we concern ourselves with.

For the type of steel that a connecting rod would likely be created with, the
ultimate tensile strength would be about 80 to 180 thousand pounds per
in^{2}. If aluminum were used, the ultimate tensile strength would be
closer to 70 thousand pounds per in^{2}. So, you can see that our
connecting rod, under a stress of 880 thousand psi, would be in serious trouble.
Failure would almost definitely occur, even if incredibly high strength steel
were used.

**REAL-WORLD RELATION**

It is in calculations like these that automotive engineers are able to
predict exactly what materials and specifications can be used in a
high-performance engine. While the data that was produced in this project
appeared to be flawed, data similar to it can be produced for each type of
engine produced. Without such knowledge, there would be a lot of guessing, and
with guessing usually comes catastrophe.

**ANOTHER SOURCE ON THE MATTER**

In an article appearing in the newsstand magazine "Engines", Jim McFarland
writes a nice article describing the most common types of failure in connecting
rods. Much of what he writes about is similar to the concepts in this project,
but he throws in a few more interesting comments.

For the most part, the type of fracture considered in this project occurred
in the center of the rod. McFarland also touches on failure at the connecting
rod bolts. These bolts attach the rods to the crankshaft journals, and
consequently, they are placed under just as much strain as the rod itself. He
states that they are often made to withstand stresses in excess of 250,000 psi.
(Which, by the way, is an awful long was from 880,000 psi, further showing how
inaccurate our data appears to be.)

As I alluded to earlier, abnormal combustion is also a serious problem for
connecting rods. McFarland notes that under detonation conditions, the pressure
from combustion can be almost double what it normally might be. Understanding
this, we can see just how susceptible a connecting rod could be to using the
wrong gasoline. (Improper octane ratings are often a causing of pinging or
knocking in an engine.)

CONCLUDING NOTES

If anything, this project should have conveyed a sense of just how
erratically an engine can operate, even under normal conditions. It is this lack
of continuity that can create major problems on parts like connecting rods.
Hence, designers and engineers are forced to choose materials that are strong
enough to withstand such powerful forces, while maintaining a low cost and
lightweight product.

Despite the fact the data in this project appeared incorrect in the final
analysis, the thought process behind it was very typical of what must be done to
analyze a complicated system. If a more accurate analysis was necessary, factors
like cylinder friction, momentum, and dozens of other variables could have been
taken into account. But, given the assumptions that were made and the data
acquired, this project still provided an interesting look at what happens inside
an engine and what limitations each engine has placed upon it.

REFERENCES

- Duffy, James E.
__Modern Automotive Technology__. 1994,
Goodheart-Wilcox Company.
- McFarland, Jim. "Connecting Rod Basics."
__Engines__. March, 1999.
Peterson Publishing Company.
- Ramos, J.I.
__Internal Combusion Engine Modeling__. 1989, Hemisphere
Publishing.
- Gere and Timoshenko.
__Mechanics of Materials: Fourth Edition__. 1997,
PWS Publishing Company.
- Beer and Johnston.
__Vector Mechanics for Engineers: Dynamics__. 1997,
WCB/McGraw-Hill.