In the previous sections, we were dealing with static analysis, which is an assumption we made
when the load is very slow and the dynamic effects of the system can be ignored.
However, in many cases, dynamic effects cannot simply be ignored and it’s even
the key part of the analysis. Like in a car crash, a drop test,
or the vibration of a rocket during launch. In a dynamic analysis we, will consider the
inertia effects of the material and additionally the effects of damping, which we will go into
detail in our next lessons So, in this roller coaster,
what is of interest to the engineer could be the forces and resulting stresses on the roller
coaster track as well as those on the car. If the car goes faster, there will generally be
more stress for both the car and the track. Let’s take another example. We have a beam with a
load on it. Imagine the load is applied slowly to the beam. Here we can use a static analysis.
But, if we suddenly drop the load onto the beam, or wish to know how that load affects
the beam during the shaking of an earthquake. All of these are dynamics problems.
So now, let’s see what’s the difference mathematically
between a dynamic and a static problem.
What extra terms or factors we need to consider for a dynamics problem?
There are three main differences that we need to address;
Inertia, Damping and loads that vary with time.
But Before we get into their details, let’s have a look at the equations of motion.
What happens if we remove the inertia, the damping, and the loads that vary with time.
We end up with the simple equation we use in a static analysis that relates the displacements
to the forces, through the stiffness matrix K. The stiffness term can be physically envisioning
as a spring. A soft body when pushed upon will deform more than a stiff body,
assuming we push with the same applied force. Bet getting back to dynamics….
Let’s have a look at inertia, From the Equations of Motion,
we see that the first term has the Mass Matrix. If you ignore all the other terms on the left
hand side of the equation, we can see the equation simplifies down to Newton's second law
which is Force = mass *acceleration. So, this mass matrix will include all
aspects of the mass and mass distribution of the object. In our rollercoaster,
if our car is heavier, it will require more force to accelerate it, which is pretty intuitive.
Essentially, inertia is the resistance of the body to any change in its velocity.
Let’s use a cyclist to illustrate this concept. A cyclist is accelerating forward from the pedaling
torque which gets translated into a Frictional Force at the rear wheel pushing the bike forward,
but aerodynamic drag from the air or wind drag and even friction from the rolling resistance
are acting in the opposite direction to the forward motion. For the cyclist to accelerate,
the Friction Force from pedaling must be greater than the sum of the drag and the
frictional rolling resistance forces. The sum of these forces will equal the inertia force.
The inertia force is sometimes called a fictitious force in the frame of
reference of the moving cyclist and acts to resist the acceleration of the cyclist,
hence we show it acting in the opposite direction, or backwards.
Likewise, a cyclist that stops pedaling and starts braking, is decelerating, and the forces acting
are the friction force on the wheels from braking, the rolling resistance as well as the wind and now
the inertia force acts in the opposite direction, resisting that change in acceleration.
Now let’s move on to the second difference we need to address.
In the equations of motion let’s look at the damping term. For this discussion we
will use this simple spring-mass system. If the spring–mass system is completely lossless,
energy would be conserved, and the mass would oscillate indefinitely.
However in real life, the motion of the spring mass system will eventually stop, because damping
is generated from the friction between air and the spring-mass system. Also, some of the energy
will go into the deformations of the spring and be transformed into small amounts of heat.
So, damping dissipate energy from a dynamic system and in our equations of motion we
see the damping force is proportional to the velocity. Think of your hand in the
water of a swimming pool. Move slowly is easy, but faster requires higher forces.
Also, in our rollercoaster example, the wheels are typically made of Urethane.
Those wheels help absorb, or dampen, the harsh vibrations of the ride.
We will cover damping in more detail in a follow on section.
Now, the third difference we need to address is the loading variation with time.
In a dynamic simulation, how a load varies with time is significant,
while in a static analysis there is no real notion of time.
Let’s use a simple diving board to illustrate this concept.
Both diving boards shown here will have a 50 pound force applied to them,
but one will be loaded quickly, while the other is loaded slowly.
When we load the board quickly, we get increased deformation, but also notice
that the board bounces in the other direction and even comes out of contact with the roller.
Interestingly, even the slowly loaded board shows some dynamic behavior as it is loaded,
as we excite some of the board's natural frequencies. How slow we can load the structure
and not induce any dynamic behavior depends on the structure. When we load slow enough
that we see little to no dynamic response, we can call this quasi-static loading.
Now, In some cases, engineers are only interested to the kinematics
or motion of the parts of a system, rather than the deformation of the materials.
For example, a robot arm assembly or a gear system.
Parts can be simplified to rigid bodies instead of continuum bodies. Such dynamics
is often called rigid body dynamics. Now clearly in our diving board example
we just discussed, we could not assume the diving board could be rigid since it is so flexible and
for it to function we need the board to bend, but in the case of a robot arm, it is meant to be very
stiff so it can be positioned extremely accurately and have limited deformation when forced.
In rigid body dynamics, the rigid bodies will not deform to force,
so we will be instead looking for quantities such as the position,
the motion, the velocity, the accelerations and the reaction forces of the parts.
Additionally, each body has 6 degrees of freedom (meaning 3 translations and 3 rotations) and
the bodies are connected through kinematic joints and even contacts.
So, relating this assumption of a rigid body back to our equations of motion, we remove the
stiffness term, you can see we are now just left with just the Mass and Damping contributions.
We’ll continue to explore dynamics in details in the following sections.