Although I'll reiterate here you DO NOT need to use the model developed
from Problem 6.1
(on the last homework) at all to complete Problem 7.1, it's worth making a few comments
about attack strategies for larger systems (described by several/many equations in a
problem statement).
Below is a block diagram for the hydraulic servomechanism described in 6.1:
You should understand both how to use/extract sub-parts of a big system and how to
create a larger system from individual sub-parts.
The first key idea that will help you is that each block of the
diagram represents some specific transfer function of its own. Various
particular groups of blocks also represent transfer functions. You may be
asked for the transfer function from e_ref to theta_t, but along the way, you
can also obtain the transfer function from (for instance) i [current] to omega_t
(angular velocity). Below is the sub-part of the total system representing the
transfer function (omega_t(s))/(i(s)): ...which can be simplified(?) to become... ("simplified" in that it's more clearly just a first-order
system)
"OK, I guess I get that idea," you say, "but how the heck would you ever figure
out the that big system at the top in the first place??"
Funny you should ask, because that's the second key idea: "a mountain is
climbed one step at a time." Yeah, that sounds cliche, but I really do think it's good
advice. Avoid panic. :^) Build up larger transfer functions from smaller sub-units.
For instance, with Problem 6.1, you can start by finding each of the following and
then linking them up. A general strategy is that after you start with some arbitrary block, you NEXT generally
solve for some block that will link up to either the input or output of what you currently have.
Then, you make some educated guess about the inertia-damping-spring system needed
to represent the rotating table and roller bearings. (I assumed just inertia and
damping here.)
This would allow you to relate the feedback transfer function from omega_t to T_t (torque at table):
...and then the relationship from T_t to P comes from a balance of POWER:
From here, we integrate each side and then reduce everything by using various relationships
given in the problem statement:
All that said, this problem was not trivial and did rely on intuition about the
spring-mass-damping characteristics of the table [you could test which elements would
result in the desired first-order response specified in the statement to figure out
if the spring and/or damping was necessary]; the POWER BALANCE step; and careful stepping
through each of the relationship given in the problem statement.
You can go through all these steps by manipulating the equations directly,
without ever using 'block diagrams', but I find think block diagrams to be a great visual aide.
