The Newton’s first law of motion states that a body will continue to be in its state of rest or
uniform moto until an external force act upon it. All the bodies, depending on their
history of loading, will undergo changes in their state until they achieve a state of equilibrium.
In some cases, this change may happen instantaneously,
or in some cases this may happen over a long time.
This depends on both the nature of the loading and the nature of the physical system.
Also, some changes may be subtle and can be ignored
while others may be too significant to be overlooked.
All these changes may affect the performance and durability of the systems
which is the motivation behind studying them over a time period. Such an analysis
where we study a system’s behavior over a time period is nothing but a time domain analysis.
But it’s not always necessary that we study a system’s behavior over a time.
So how do we decide a need for a time domain analysis
and how do we assess the time range over which we need to analyze a system’s behavior?
This boils down to the physics that’s related to the said changes.
From an engineering standpoint, sometimes we’re only interested in the system’s behavior
after it achieves a state of equilibrium and sometimes,
we’re interested to know what happens to it while it is achieving this state of equilibrium.
Either way, it is important for a good engineer to identify the physical phenomena
that are changing a system’s behavior and then study them for an appropriate amount of time.
In this section we’ll discuss the primary reasons that bring about a time-dependent behavior
in mechanical systems that an engineer must consider in their design process.
We’ll focus on three aspects namely 1. Time varying loads and boundary conditions
2. Time dependent material behavior 3. And inertial and damping effects
Let’s look at few examples of each of these types to better understand how they make
mechanical system time dependent and how a time domain analysis can help us analyzing them.
We’ll start with a very common occurrence which is time varying loads and boundary conditions. We see
this everywhere and every day. For instance, if we place our smart phone on the ground
then the weight of the phone is the only load that is acting on it and it is acting gently so
it wouldn’t do any damage to the phone. However, if we drop the same phone from a certain height,
upon its impact with the ground, its weight is still the only load that is acting on it but it
acts rather abruptly, or over a very short period of time and that can potentially damage the phone.
In this scenario, the time over which the same amount of force acts makes a huge difference in
how the phone device reacts to it. This is a classic example of time varying load and
a time domain analysis will help us figure out whether the drop would damage the phone.
Similarly, any physical application that involves transmission of a wave in a media
such as acoustic applications is also a time dependent system.
Time-dependent loads are also used in designing complicated machinery.
A very good example is a cam shaft that has different cams along its length that may be
used to time the opening and closure of the valves. Usually, as the number of cams increase
it gets very complicated to keep track of the timing profiles. In such a case a time
domain analysis can come in very handy in designing complex mechanical systems.
There are many more such examples for time varying loads
but let’s move on to the next category which is the time dependent material behavior.
Unlike the earlier case, this aspect of time dependence
is coming from the inherent system property which is the material behavior.
Some materials have a noticeable time dependent behavior which results in two types of behavior:
1. Stress relaxation and 2. Creep deformation
These two phenomena are modeled using advanced techniques
which are out of the scope for this discussion but let’s take a quick look at their definitions.
Stress relaxation, as the name suggests, is a decrease in the
stress developed in the body with time when it is subjected to a constant amount of strain.
So, this phenomenon should be taken in account when a system is designed to operate
at a certain stress level over time. A good example for this scenario is bottle cap,
a certain capping pressure depending on the product contained in that bottle
is generally preferred. Usually the bottle caps are made up on polymer plastics which
are known to undergo stress relaxation. So, it is preferred that during the bottling process
the cap is tightened to a capping pressure slightly more than the preferred value,
so it won’t decrease below the preferred value during its shelf life.
In this case a time domain analysis is very useful in estimating how large the capping pressure must
be so that the product would survive the shelf life during which it undergoes stress relaxation.
The second type of time dependent material behavior is creep which is an increase in the
strain in the body with time when the material is subjected to a constant amount of load.
A good example where this is seen is in electronic equipment
in which different components are soldered in place. The solder material in general is
made of lead which is known to undergo creep deformation especially at high temperatures.
A failure of the solder can result in failure of the electronics due to loss of connection.
So, the time dependent creep behavior of solder material needs to be considered while assessing
the reliability of electronic equipment and a time-domain analysis is helpful in doing so.
Now let’s look at the third category of time dependency which is the inertial and damping
effects. These two phenomena are dependent on both the system properties and the external loads.
Inertia is a property of matter to stay in its state of rest of uniform motion unless
an external force is applied. This depends on the mass of the system,
how is it distributed in the system and how fast is it moving in space.
All these factors contribute to the inertial effects that a system experience.
A time domain analysis is needed to study how a system reacts under inertial loads.
This is an important factor in analyzing mechanics of
occupants during crashworthiness, or vibrating structures or any other dynamic system.
In terms of energy, the inertia of a body is responsible for storing the kinetic energy in it.
Unlike inertia, damping converts kinetic energy of a system into heat energy and dissipates it.
This way it reduces the tendency of a structure to move. This phenomenon reduces the amplitude
of a vibrating system over time and therefore a time-domain analysis is necessary to study.
In the current lesson we will learn about inertia in detail,
but we’ll save the discussion on damping for a future course as it is a more advanced topic.