We have gained some insight into what time domain analysis is and how is it useful
in understanding various mechanical systems. With this in mind let’s now tackle a few
real applications and see how time domain analysis is used in those cases.
We have lined up four different examples each of which highlights different reasons for time
dependency in the model. These four examples are: Billiards game
A cam shaft analysis The shelf-life analysis of a bottle cap
And a cool toy that gained popularity very recently – a fidget spinner!
Now let’s dive into these applications see how time domain analysis is applicable to them.
Let’s start with billiards or snooker or pool, a very interesting game that dates to 15th century
when it was first played as an outdoor game and slowly made its way to indoor pool table.
Today there are various versions of this game but a common theme among all the versions
is to knock the billiards balls into the hole by hitting the cue ball with a cue stick.
It’s a very simple game, all one has to do is to have control over the speed of the cue stick,
the angle of impact with the cue ball, the angle of impact both between the
balls and also the ball and the wall and a few other factors. Sounds easy, right?
Well, every player makes decisions about all these factors while playing and
the person making better judgement has the best control over their shots.
Now let’s dissect the mechanics of this problem,
when a player hits a cue ball with a cue stick, they transfer the momentum into it. The cue ball
then makes an impact with the other balls and in that process transfer the momentum to them.
The angle at which it hits different balls, the speed with which it hits them and any spin
that it has about its own axis will all affect the way the balls are scattered on the board.
So, if you notice, the kinetic energy of the cue ball is transferred to rest of the balls
upon collision and this energy exchanges happen repeatedly whenever there’s a collision.
Since there is a time evolving kinetic energy happening, this system requires
a time domain analysis and would require full set of equations to be solved.
Let’s simulate the start of the game and monitor the velocity of the cue ball
and one of the billiards ball and plot it over time we can see from the response how the cue
ball slows down and at the same instant the billiards balls picks up the speed.
We can notice a similar behavior by monitoring their kinetic energies as well.
Also, this exchange of energy happens over a fraction of a section
so this can be classified as a time varying load problem.
Now, let’s move on to another time varying load problem which is more inline with an industrial
application. A cam shaft is a mechanical component that’s popularly used for timing
the opening and closure of valves. Camshafts have been employed since 1200 BC and are used
eve today in internal combustion engines to time the opening and closure of valves.
In general, the design of a cam shaft includes a series of cams with certain
profile mounted along the length of a shat which is rotated at a designated speed.
As they rotate, depending on the profile and the position of the cam, different
valves remain open at a certain desired time. Designing a cam shaft can be a tricky business,
as the number of cams increase keeping track of timing can become very complicated.
A time domain analysis can come in very handy for such cases. Let’s simulate the
function of a cam shaft that has four cams at different positions and at different angles.
We will use a time domain analysis and monitor the displacement of each valve
over time to see when they open and close. Ideally, we would want only one valve
open at a time in order to maintain efficient operation of the engine.
From the animation we can see that not all valves are open at the same time.
But let’s look at the displacement profile of each valve
and see if there’s an overlap between the opening of any two valves.
From the plots we can clearly see that no two valves are open at the same time.
Also, we can see the full length of time for which each valve remains open.
This again shows how a time domain analysis can be useful in designing machine components.
Now, let’s shift our focus to the next type of time dependency
which is the time dependent material. For this demo we’ll look at the shelf life of a bottle cap.
Plastic bottles are used as containers for many products ranging from laundry liquid to sodas.
Generally, these bottles and caps are made of different polymer plastic
due to their low weight and high strength. During packaging, a bottle cap is tightened to until a
certain amount of contact pressure is established due to the deformation of the bottle and cap.
The value of this contact pressure depends on the contents of the bottle, for instance the
pressure in case of a laundry detergent should be large enough o prevent any leakages but in
case of a soda bottle this should be higher to make sure that the drink does not lose its fizz.
Once the package is done, these bottles typically have a shelf life of, say,
3 weeks before they’re opened by a consumer. But polymer plastics are known to undergo stress
relaxation so in this 3 week of shelf life the contact pressure would drop down and as a result
it may decrease below the desired pressure and in turn affect the quality of its contents.
So, to fix this issue, the capping pressure at the time of packaging is set to a higher value than
the desired value so that it wouldn’t drop lower than the desired value during its shelf life.
Deciding the value of this capping pressure is not straight forward and it depends on the material
used for the bottle-cap assembly, so a time domain analysis is used in estimating the value.
In this example we have a simplified bottle-cap assembly
and we generate a contact pressure by tightening the cap.
Then we let it sit on the shelf and monitor the contact pressure over time.
As we can see it drops to a certain value over its shelf life.
This behavior can be captured only if we include the relevant material behavior,
which is viscoelasticity in this case.
If we do not account for this time dependency, then we will never be able to predict the drop-in
contact pressure over the time which can incur a lot of business losses.
Finally, let’s move on to a new sensation of a toy the fidget spinner. It’s a cool mechanism
where we hold the spinner with two fingers in the center and when we
give it a spin, it continues to rotate. Its design generally includes two or three lobes
with each carrying a certain weight rotating about a bearing in its center pad. In general,
the spinner body is made of some form of plastic, but the lobes contain a metal weight.
This is because, the fidget spinner uses the concept of inertia, so when we give it a spin,
the energy we impart is stored as kinetic energy by these weights and since there’s a minimal loss
of energy due to friction at the bearings it continues to rotate for a long time.
Now let’s design one such fidget spinners and use simulation for this purpose.
If we hold the spinner fixed in the center and impart a moment t the end,
if the lobes contain weights made of structural steel,
it starts to spin at an angular velocity of close to 700 rpm.
Now let’s replace the steel lobes with a lighter metal such as aluminum and see
how fast it spins for the same moment load. From the simulation we see that it rotates at
an angular velocity of about 130 rpm which is over 5 times less than steel lobes.
This is because steel lobes are heavier
and can carry more kinetic energy due to inertia compared to aluminum lobes.
A flywheel operates on the same concept, a typical example is seen in potter’s wheel
which has been in use since 3rd century BC. Even today we use flywheel energy storage systems
in various sophisticated assemblies all of which operate on the concept of inertia.
Since the inertial terms are carried by the mass terms,
a time domain analysis is necessary to study and design such systems.
With these examples we can conclude our discussion on time domain analysis.
Depending on the length of the time period of interest there are different solution techniques
which can be used in analyzing these systems. We will discuss such methods in a future course,
for now understanding the physics behind these examples is the scope of the current lesson.