- [Instructor] In this video, we're going to look at
an introduction to normal modes analysis.
And the very first question is,
what is normal modes analysis?
Now we find there's quite a few terms used
to describe the same thing.
We find we've called it eigenvector analysis,
eigenvalue analysis.
SOLIDWORKS calls this a frequency analysis.
The terminology can be very confusing.
One way to clear up the confusion
is to look at the objectives of the analysis.
The first objective is to find all the resonant
or natural frequencies within a range.
And these are defined in cycles per second or hertz,
or occasionally in radians per seconds or rad per second.
The second objective is to find the mode shapes
at each natural frequency.
Now these are only shapes.
There's no actual unit involved, and there's no amplitude.
Because we're not applying a force input,
so we can't get a scaled response coming out.
We saw in the previous video
that the full equation of motion was defined
by an internal energy balance.
We do a similar kind of thing for the natural frequencies.
The first form of energy is the kinetic energy,
created by accelerating the mass.
Then we have potential energy, which is stored energy
created by the deflection of the structure.
And now we ignore the external loading,
and we ignore damping.
Now the equation of motion is greatly simplified.
We've knocked out the damping force,
and we've knocked out the applied force.
We're just left with the inertia force
and the stiffness force.
So the equation of motion is now
a simple energy balance relationship.
A key assumption in solving this equation
is to assume simple harmonic motion.
That means basic combinations of sines and cosines.
These are described in terms of a harmonic frequency,
omega n.
This is also described as the natural frequency.
Over a single degree of freedom system,
the expression's given by square root of k over m.
Omega n is described in terms of radians per second,
but we can also describe it as a natural frequency
in terms of hertz.
The relationship square root k over m
is a very important equation.
If we increase the stiffness,
then we increase the natural frequency.
If we increase the mass, we reduce the natural frequency.
For a multiple degree of freedom system,
the natural frequency for a particular mode
is given by the distribution of effective stiffness
and effective mass.
For a system in multiple degrees of freedom,
an equation of motion becomes a matrix equation.
So we have a mass matrix, we have a stiffness matrix,
and we have an eigenvalue solution,
whose form is shown here, with the eigenvalue
proportional to the natural frequency squared,
and the eigenvector is the mode shape.
Now theoretically, if we have n degrees of freedom
in a dynamics problem, that can produce n eigenvalues,
n natural frequencies, and n corresponding eigenvectors
or mode shapes.
So theoretically, if I had 1,000,000 degrees of freedom
in a model, I could produce 1,000,000 natural frequencies.
Luckily for us in practice,
we only need a small number of modes.
The largest number of modes I've ever used is about 500,
and sometimes we only need between 10 and 15 modes.
The fundamental reason is that it takes more and more energy
to drive higher and higher frequency modes,
so most excitation energy is generated by the lower modes.
But we do have to keep an eye
on higher frequency driving inputs,
and make sure we have enough modes to capture
up to that frequency range.