- [Instructor] In this video we're going to look
at the background to damping.
Damping is present in all oscillatory systems,
so all of the FEA models we we're using
for Dynamics should include damping.
Damping always acts to dissipate energy out of a system.
In the real-world damping mechanisms are very complicated,
they're difficult to predict analytically,
and they're difficult to measure experimentally.
We make a series of very simple damping
assumptions in Linear Dynamics.
Probably the most important is that we assume
damping is viscus as proportional to velocity.
In transient response analysis
or frequency response analysis, we must include damping,
but for normal modes analysis we ignore damping.
As we've said, the damping effect is to dissipate energy,
in other words energy out.
The picture here shows a short durational loading,
in fact it's an initial displacement condition
which is then let go.
In that particular situation,
the energy out or the energy dissipated
decays the response down to zero.
In contrast to that, we now how a harmonic loading.
Which is, for example, signal sorting loading,
which is on throughout the analysis.
So energy is continuously being pumped into the system.
Now initially, damping hasn't had a chance
to build up and isn't dissipating very much energy.
Gradually the dissipation level increases
and at the steady state condition
the energy being driven in is balanced
by the energy being dissipated by damping.
So the response rises in the transitory period
and then stays flat in the steady state period
where energy in balances energy out.
When we talk about a damping level,
that's usually defined as a percentage of critical damping.
On the graph on the left,
critical damping is occurring at 100%
and that's a level of damping where
there's just zero oscillation.
The structure will return to the zero condition
in the fastest possible time.
Underdamped levels are less than 100%
and oscillations will occur and this is the typical
situation when we're doing a dynamic analysis.
Overdamped is where the damping is bigger than 100%
and the return time is slower than the critical damping.
Again, no oscillations will be occurring in that situation.
The typical values of damping we would see
in linear dynamics vary anywhere between,
say a half a percent to heavier damping with around 5%.
Several alternative assumptions are made about damping
to help the calculation process.
The first big assumption is that damping
can be considered as modal.
This means that each mode can potentially
have a different damping level.
Now the damping only acts
on that particular modal contribution.
This form of damping acts to decouple modal response.
So if we're using modal methods, then it's very convenient
because it will decouple down into single
degree of freedom equivalent problems.
Then again modal damping is a viscus
form of damping assumption.
An alternative assumption to modal damping
is Rayleigh damping.
We can use modal damping or Rayleigh damping
in modal based solutions, but for direct transient,
then we're forced to use Rayleigh damping.
The form of the Rayleigh damping is shown here.
The damping matrix is proportional to the Mass matrix
in the coefficient Alpha are proportional
to the stiffness matrix and the coefficient Beta,
so it's called Mass proportional
or Stiffness proportional damping.
The form of the curve is shown in the top figure.
The net result is to calculate percentage critical damping
as a function of frequency.
Now Rayleigh damping is very error prone.
It's very easy to get the coefficients wrong
and they don't really give any insight
into the form of the damping,
so I strongly recommend plotting the damping out in excel.
The damping level is a percentage
critical against the frequency.
If you use this form of the equation
then we know Alpha and Beta.
So for every frequency you can calculate
what the percent of critical damping is.
If you want to estimate Alpha and Beta for yourself,
rather than take it from standard values,
then a typical approach would be
to pick two dominant frequencies
and the levels of damping you want
and then you have two equations and two unknowns,
so you can recast the equation on the left twice.
Once with no unnatural frequency,
once with a second natural frequency.
Then you can solve for the coefficients
and then again my recommendation is to plot that
on the graph, just to see where the other frequencies lie.
You may find is that you match the two frequencies
to your target, but then you may have
a very high or very low damping
in other important frequencies.
Getting the damping level right is very important.
Particularly, if we're driving or exciting
close to the resonant frequencies.
For short duration excitation
or if we're driving well away from natural frequencies,
then the damping is not quite so critical.
Damping is quite tricky to set up,
so it's a good idea to take some care with it.
Also to check the levels of damping
which are coming out of the analysis at the other end.