LAB #6 report. MAE 106. UCI. Winter 2005

Nasser Abbasi, LAB time: Thursday 2/17/2005 6 PM

Answer 1.

part(a)

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:

part(b)

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

part(c)

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

Answer 2.

part(a)

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

part (b)

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.

Part (c)

Hence

Hence the natural frequency for the linear (translation) motion is