- [Instructor] In this video we're going to look at
the foundations of Modal and Direct response methods.
We're going to look at the first four mode shapes,
or Eigenvectors, of the plate.
Here's mode one, mode two,
mode three, mode four.
Notice how I've used the expression,
for the set of Eigenvectors,
phi one, phi two, phi three and phi four.
Now a normal modes analysis can store these mode shapes,
or in other words, these Eigenvectors.
And there's a schematic of the storage
where the Eigenvector Matrix, phi,
is a collection of all the Eigenvectors.
This collection of Eigenvectors
is described as the Modal Database.
We can use these stored Eigenvectors
in a Modal response analysis.
Let's look at a Transient response excitation of the plate.
It's loaded at the corner node
and we see a typical response at some snapshot in time.
The actual time is .097 seconds.
The complete transient time history
of the corner node is shown,
and also put a vertical line at .097 seconds.
If we're able to track an animation of the tip through time,
it would be exactly the same as a tip deflection time
history you see here.
Now at this snapshot in time,
the Eigenvectors can be scaled and then combined together
to give the total physical response.
The mode shapes are always the mix that go
to make the total response of the structure.
So here's mode one with some scaling factor Q,
or Q one, being applied to it.
Mode two, the torsional mode,
also has a scaling factor of Q two applied.
Mode three similarly has a separate scaling factor
Q three applied to it.
And finally, mode four, also has its own
independent scaling factor applied to it.
Now these are the modes which go to make up
the mix with these specific scaling factors being applied.
So, this is the equation form,
the physical Degrees of Freedom, X,
are equal to the Matrix of Eigenvectors
times the scaling factors.
And the scaling factors are in fact called
the Modal Coordinates.
That's the basis of the Modal Method.
And just to see how that multiplication is carried out
in full, here's the transformation expression
from the physical coordinates, the Modal coordinates,
and you can see it is literally
just a row times column scaling on each Eigenvector
scaled by its corresponding Modal coordinate.
So the approach we take in the Modal Method
is first off we find the natural frequencies
and normal modes,
in other words, the Eigenvectors,
we save the Eigenvectors into the Eigenvector database,
we define the response analysis in Modal coordinates,
and Modal coordinates are Q,
and then we solve the response analysis
in the Modal coordinates.
And we transform the solution back to physical coordinates
using the transformation that we've seen.
Now the big motivation for using Modal coordinates
is that the Transient and Frequency Response analyses
decouple into a set of Single Degree of Freedom solutions.
So, if we have five Modal coordinates,
we have five separate Single Degree of Freedom equations.
There is one solution per mode.
What it means is that large physical dynamics problems
can be reduced to very small modal dynamics problems.
So, for example, a million Degrees of Freedom,
if we've only got 20 modes, then it just becomes
a 20 Degree of Freedom problem to solve.
The advantages of the Modal Method
are clearly we're gonna get
a significant CPU time reduction.
We can also run a response in two steps:
first off run the Modal analysis to generate
the Modal Database, and then use that Modal information
to carry out the response analysis.
The Modal Database allows very fast re-run
the Eigenvectors are already sitting there,
waiting to be used.
We can also use Modal filtering.
And in the various examples I use that quite a lot
to show how we can break down,
and essentially tune out, particular frequencies.
For example, we can run with just a mode one
to see what the effect is.
Or we can run just with a mode five present
to see what the high frequency content is,
which otherwise might be hidden.
One big advantage of the Modal frequency response
that we'll see coming up shortly,
is that it inherits Natural Frequencies.
So the Natural Frequencies are known,
what we're going to describe as the Peak Points
are therefore known automatically.
And just to expand on the Modal Filtering,
if we go to a Transient response excitation
of the plate, for example,
this is the full acceleration time history coming through
of the corner node.
And, as it's the full response,
by definition it includes all six modes.
This is the response just with mode one,
so we've filtered everything out other than mode one.
Mode one is the first bending mode: low frequency.
Now I've run again and this time I've filtered out
everything except for mode two.
Mode two is a higher frequency,
we can see that from the graph,
and it's a torsional mode.
Mode three is now shown filtered.
We can see again by eye a very much higher frequency.
The actual magnitudes are also decreasing.
The scale is reducing as I plot the higher modes.
A mode three is the bending two mode: second order bending.
Finally, we have mode four, which I've filtered,
and you can see very clearly the effect of damping there.
It's a nice exponential decay.
And mode four is a torsion two mode:
second order torsional mode.
Now SOLIDWORKS Simulation uses Modal and Direct Methods
in a very specific way.
Other solvers use different combinations.
In SOLIDWORKS Simulation,
Frequency Response is a modal solution only,
there's no option to use a Direct Method.
One the face of it, Linear Transient Response
is also modal only, but there's a but there.
Nonlinear Transient Response is by definition
a Direct Only solution,
but we can remove all the nonlinearity,
so effectively running a Linear Transient,
so we can run a Direct Linear Transient.
And there are good reasons why we would
sometimes want to run a Direct Linear Transient.
So what are the advantages of running the Direct Transient?
The big advantage is that the Direct Transient
doesn't have any modal truncation.
That's a big problem with Modal Transients.
If we don't specify sufficient modes,
we literally clip the high-frequency content.
With the Direct Method, it will try and find
all the modes it can
based on the fidelity of the time step.
If we have a very fine time step,
then it can find high-frequency content.
On the other hand, if we use a very coarse time step,
then it won't have any accuracy associated
with the higher frequency content.
What I normally do is develop a model using Modal Methods,
and then I will compare the final result
with a check Direct Transient.
The disadvantages of a Direct Transient
are that it's more expensive in CPU,
and we can't use any of the modal filtering methods.
Which is why I usually use a two-pronged approach:
Modal Methods to de-bug and understand the model,
and then the Direct Method just to complete the picture
and make sure I haven't made any Modal Truncation.
The other disadvantage with the Direct Method
is that we have to use Rayleigh Damping,
which I consider to be a rather clumsy form of damping.
Unfortunately in SOLIDWORKS we don't have the option
to have a structural damping,
which is proportional to stiffness.
So, in summary, we've looked
at Modal Methods and Direct Methods,
and I think the key thing is to understand,
at least in concept, what a Modal Method is doing,
and how you can take advantage of it
with filtering techniques, and so on.