LAB #6 report. MAE 106. UCI. Winter 2005
Nasser Abbasi, LAB time: Thursday 2/17/2005 6 PM
Free diagram for model (2) is the following (assuming m1 is moving to right
faster than
m2)
Now
derive equations. Take right to be positive.
For m1:
For m2:
determine transfer function
Write the dynamic equations in matrix form, we get
Above can be written as
Hence , taking laplace transform we get
Now,
Hence
Now
But for the above,
Hence
i.e.
and
Let
hence
will not move
when
but
implies
.
i.e.
.
But this is the sum of 2 positive quantities. So it is only possible to sum to
zero only when each quantity itself is zero. i.e.
But for non zero
this means that
.
But
(the samping) is not zero, since we do have damping in the systems, hence it
is not possible
that
. In otherwords, there will not be an isolation fequency, and
will always be non-zero.
But if
is very small, then
and in this case
when
or when
Need to derive a mathematical model. First step is to make a block diagram as
follows.
There are 2 motions. One rotational about the center of mass, and one translation, up and down.
Free body diagrams
are
Now the equation of motion for the rotational motion is
But
Hence we get for small
,
using
For the translation motion,
hence
Write the above in matrix form, we get
Take laplace transform we get
Let
Hence above matrix equation can be written as
Take laplace transform, we get
where
is the
identity matrix.
let
we get
multiply both side by
we get
i.e.
Which is what we are required to show.
Hence
Hence the natural frequency for the linear (translation) motion is