At
left a spring/mass/dashpot system is shown, driven by a piston at the
top.
The
equation governing this system is displayed in yellow at the top. Mass
is set to 1; b is the damping constant, k is the spring constant, and
ω is the circular frequency of the sinusoidal motion of the piston.
To
the right, the position of the plunger (the input signal cos(ωt))
is graphed in cyan, and the position of the mass (the system response
x ) is graphed in yellow. Diamonds indicate the current values cos(ωt)
and of x, and a vertical white line between them indicates the extension
of the spring. A grey vertical line measured by a red segment indicates
the time lag t0 (which is also read out in red at the bottom
of the screen, below a readout of the period P in cyan).
Rolling
the cursor over the graphing window produces crosshairs and a readout
of the values of t and x.
The time value is set using a slider under the window. The [>>] key starts an
animation. The [<<] key resets t to
t = 0.
Grab
the [b], [k], or [ω] slider to vary those parameters.
The
[Bode plots] key toggles display of two windows on the right side of
the screen. The top window displays the amplitude A of the sinusodial
response as a function of ω. The window below it displays the
negative of the phase lag φ as a function of ω.
The
[Nyquist plot] key toggles display of a window at lower right, showing
a portion of the complex plane. On it, a grey curve traces the path
traversed by the complex gain (k+ biω) / p(iω) (where p(s)
= s2 + bs + k is the characteristic polynomial) as ω
varies over positive values. A yellow diamond marks the value of this
complex number for the chosen value of ω. A yellow line segment
connects it to the origin. The length of this segment is the amplitude
A, and the angle up from the positive real axis, marked by a a green
arc, is -φ.
Roll
the cursor over the amplitude window to cause a horiziontal yellow line
to appear relating the amplitudes in the three top windows, along with
a readout of the amplitude. Roll the cursor over the phase shift window
to cause a readout of the phase shift.
Note:
These are not quite truly Bode or Nyquist plots. A Bode plot graphs
log(A) vs log(ω) or -φ vs log(ω). A Nyquist plot displays
k/p(i ω) as omega ranges from -∞ to +∞ it has a portion
above the real axis which is symmetric with what is drawn.
©
2001 H. Hohn