Gain and Phase
Shift
Part 2: Finding Gain
and Phase Shift
We now think of our driving
function as "tunable" -- that is, we may vary the input frequency
omega, but the system parameters c and k remain fixed.
Thus, gain and phase shift are functions of omega given by the formulas
and
In your worksheet you will
find commands for plotting the solution of a damped linear oscillator driven
by a sinusoidal input with frequency omega = 0.5. The input function
is also graphed.
- By direct measurements
on the graphs, determine the amplitude of the steady state response and
the gain M(0.5).
- By direct measurements
on the graphs, determine the phase shift phi(0.5).
- Vary the initial conditions,
and re-solve the differential equation. Repeat steps 1 and 2 to confirm
that your results do not depend on the initial conditions.
- As you did in the preceding
steps, determine M(omega) and phi(omega) for omega = 0.25,
0.75, 1.0, 1.5, 2.0, 3.0.
- Sketch the graphs of M(omega)
and phi(omega) directly from the data you have gathered.
