Assignment 3.pdf - UBC Mechanical Engineering

l
v
= = the lever arm ratio of the handle rocker
lh
m = equivalent lumped mass of the valve flapper and the lift rod
k = stiffness of spring action on the valve flapper.
The damping force f NLD on the valve is assumed quadratic and is given by
f = av v
NLD VLD VLD
where, the positive parameter:
a = a u for upward motion of the flapper (v NLD ≥ 0)
= a d for downward motion of the flapper (v NLD < 0)
with a u >> a d
The force applied at the handle is f(t), as shown.
We are interested in studying the dynamic response of the flapper valve. Specially, the valve
displacement x and the valve speed v are considered outputs, as shown in Figure P4.10. Note
that x is measured from the static equilibrium point of the spring where the weight mg is
balanced by the spring force.
(a) By defining appropriate through variables and across variables, draw a linear graph
(b)
for the system shown in Figure P4.10. Clearly indicate the power flow arrows.
Using valve speed and the spring force as the state variables, develop a (nonlinear)
state-space model for the system, with the aid of the linear graph. Specifically, start
with all the constitutive, continuity, and compatibility equations, and eliminate the
auxiliary variables systematically, to obtain the state-space model.
(c) Linearize the state-space model about an operating point where the valve speed is v .
For the linearized model, obtain the model matrices A, B, C, and D, in the usual
notation. The incremental variables ˆx and ˆv are the outputs in the linear model, and
(d)
the incremental variable f ˆ( t ) is the input.
From the linearized state-space model, derive the input-output model (differential
equation) relating f ˆ( t ) and ˆx .
Problem 11 (Problem 4.11 from Textbook)
A common application of dc motors is in accurate positioning of a mechanical load. A schematic
diagram of a possible arrangement is shown in Figure P4.11. The actuator of the system is an
armature-controlled dc motor. The moment of inertia of its rotor is J r and the angular speed is
ω r . The mechanical damping of the motor (including that of its bearings) is neglected in
comparison to that of the load.
The armature circuit is also shown in Figure P4.11, which indicates a back e.m.f. v b (due
to the motor rotation in the stator field), a leakage inductance L a , and a resistance R a . The
current through the leakage inductor is i L . The input signal is the armature voltage v a (t) as shown.
The interaction of the rotor magnetic field and the stator magnetic field (Note: the rotor field